Chapter 8 Resource Masters
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Contents
Teacher’s Guide to Using the Chapter 8
Resource Masters ........................................... iv
Lesson 8-5
Student-Built Glossary ....................................... 1
Anticipation Guide (English) .............................. 3
Anticipation Guide (Spanish) ............................. 4
Dot and Cross Products of Vectors in Space
Study Guide and Intervention .......................... 26
Practice............................................................ 28
Word Problem Practice ................................... 29
Enrichment ...................................................... 30
Graphing Calculator Activity ............................ 31
Lesson 8-1
Assessment
Chapter Resources
Introduction to Vectors
Study Guide and Intervention ............................ 5
Practice.............................................................. 7
Word Problem Practice ..................................... 8
Enrichment ........................................................ 9
Vectors in the Coordinate Plane
Study Guide and Intervention .......................... 10
Practice............................................................ 12
Word Problem Practice ................................... 13
Enrichment ...................................................... 14
Chapter 8 Quizzes 1 and 2 ............................. 33
Chapter 8 Quizzes 3 and 4 ............................. 34
Chapter 8 Mid-Chapter Test ............................ 35
Chapter 8 Vocabulary Test ............................. 36
Chapter 8 Test, Form 1 ................................... 37
Chapter 8 Test, Form 2A................................. 39
Chapter 8 Test, Form 2B................................. 41
Chapter 8 Test, Form 2C ................................ 43
Chapter 8 Test, Form 2D ................................ 45
Chapter 8 Test, Form 3 ................................... 47
Chapter 8 Extended-Response Test ............... 49
Standardized Test Practice ............................. 50
Lesson 8-3
Answers ........................................... A1–A23
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Lesson 8-2
Dot Products and Vector Projections
Study Guide and Intervention .......................... 15
Practice............................................................ 17
Word Problem Practice ................................... 18
Enrichment ...................................................... 19
Lesson 8-4
Vectors in Three-Dimensional Space
Study Guide and Intervention .......................... 20
Practice............................................................ 22
Word Problem Practice ................................... 23
Enrichment ...................................................... 24
Graphing Calculator Activity ............................ 25
Chapter 8
iii
Glencoe Precalculus
Teacher’s Guide to Using the
Chapter 8 Resource Masters
The Chapter 8 Resource Masters includes the core materials needed for Chapter 8. These
materials include worksheets, extensions, and assessment options. The answers for these
pages appear at the back of this booklet.
Practice This master closely follows the
types of problems found in the Exercises
section of the Student Edition and includes
word problems. Use as an additional
practice option or as homework for
second-day teaching of the lesson.
Chapter Resources
Student-Built Glossary (pages 1–2) These
masters are a student study tool that
presents up to twenty of the key vocabulary
terms from the chapter. Students are to
record definitions and/or examples for each
term. You may suggest that students
highlight or star the terms with which they
are not familiar. Give this to students before
beginning Lesson 8-1. Encourage them to
add these pages to their mathematics study
notebooks. Remind them to complete the
appropriate words as they study each lesson.
Word Problem Practice This master
includes additional practice in solving word
problems that apply to the concepts of the
lesson. Use as an additional practice or as
homework for second-day teaching of
the lesson.
Enrichment These activities may extend
the concepts of the lesson, offer an historical
or multicultural look at the concepts, or
widen students’ perspectives on the
mathematics they are learning. They are
written for use with all levels of students.
Graphing Calculator, TI–Nspire
Calculator, or Spreadsheet
Activities These activities present ways in
which technology can be used with the
concepts in some lessons of this chapter. Use
as an alternative approach to some concepts
or as an integral part of your lesson
presentation.
Lesson Resources
Study Guide and Intervention These
masters provide vocabulary, key concepts,
additional worked-out examples and Guided
Practice exercises to use as a reteaching
activity. It can also be used in conjunction
with the Student Edition as an instructional
tool for students who have been absent.
Chapter 8
iv
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Anticipation Guide (pages 3–4) This
master, presented in both English and
Spanish, is a survey used before beginning
the chapter to pinpoint what students may
or may not know about the concepts in the
chapter. Students will revisit this survey
after they complete the chapter to see if
their perceptions have changed.
Leveled Chapter Tests
Assessment Options
The assessment masters in the Chapter 8
Resource Masters offer a wide range of
assessment tools for formative (monitoring)
assessment and summative (final)
assessment.
• Form 1 contains multiple-choice questions
and is intended for use with below grade
level students.
• Forms 2A and 2B contain multiple-choice
questions aimed at on grade level
students. These tests are similar in
format to offer comparable testing
situations.
• Forms 2C and 2D contain free-response
questions aimed at on grade level
students. These tests are similar in
format to offer comparable testing
situations.
• Form 3 is a free-response test for use
with above grade level students.
All of the above mentioned tests include a
free-response Bonus question.
Quizzes Four free-response quizzes offer
assessment at appropriate intervals in
the chapter.
Mid-Chapter Test This one-page test
provides an option to assess the first half of
the chapter. It parallels the timing of the
Mid-Chapter Quiz in the Student Edition
and includes both multiple-choice and
free-response questions.
Vocabulary Test This test is suitable for
all students. It includes a list of vocabulary
words and questions to assess students’
knowledge of those words. This can also be
used in conjunction with one of the leveled
chapter tests.
Extended-Response Test Performance
assessment tasks are suitable for all
students. Sample answers are included for
evaluation.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Standardized Test Practice These
three pages are cumulative in nature. It
includes two parts: multiple-choice questions
with bubble-in answer format and
short-answer free-response questions.
Answers
• The answers for the Anticipation Guide
and Lesson Resources are provided as
reduced pages.
• Full-size answer keys are provided for the
assessment masters.
Chapter 8
v
Glencoe Precalculus
NAME
DATE
8
PERIOD
This is an alphabetical list of key vocabulary terms you will learn in Chapter 8.
As you study this chapter, complete each term’s definition or description.
Remember to add the page number where you found the term. Add these pages to
your Precalculus Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term
Found
on Page
Definition/Description/Example
component form
components
cross product
direction
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
dot product
equivalent vectors
initial point
magnitude
opposite vectors
ordered triple
(continued on the next page)
Chapter 8
1
Glencoe Precalculus
Chapter Resources
Student-Built Glossary
NAME
DATE
8
PERIOD
Student-Built Glossary
Vocabulary Term
Found
on Page
Definition/Description/Example
orthogonal
parallel vectors
quadrant bearing
resultant
standard position
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
terminal point
true bearing
unit vector
vector
z-axis
Chapter 8
2
Glencoe Precalculus
NAME
8
DATE
PERIOD
Anticipation Guide
Step 1
Before you begin Chapter 8
• Read each statement.
• Decide whether you Agree (A) or Disagree (D) with the statement.
• Write A or D in the first column OR if you are not sure whether you agree or disagree,
write NS (Not Sure).
STEP 1
A, D, or NS
STEP 2
A or D
Statement
1. Scalars have both magnitude and direction.
2. Vectors can be represented by directed line segments.
3. The initial point of a vector is the point where the vector
starts.
4. Vectors can be used to represent forces applied at an angle.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. When vectors are combined the result is a scalar.
6. Any vector can be broken down into horizontal and vertical
components.
7. To add two vectors algebraically, add all of the numbers in
the first vector together. Then add all of the numbers in the
second vector together.
8. Trigonometric ratios sometimes need to be used when
working with vectors.
9. The dot product of two vectors is a scalar.
10. Vectors can be used to represent forces in three-dimensional
space.
11. When finding torque, the dot product is used.
12. The area of a parallelogram can be found using dot products.
Step 2
After you complete Chapter 8
• Reread each statement and complete the last column by entering an A or a D.
• Did any of your opinions about the statements change from the first column?
• For those statements that you mark with a D, use a piece of paper to write an example
of why you disagree.
Chapter 8
3
Glencoe Precalculus
Chapter Resources
Vectors
NOMBRE
8
FECHA
PERÍODO
Ejercicios preparatorios
Identidades y ecuaciones trigonométricas
Paso 1
Antes de que comiences el Capítulo 8
• Lee cada enunciado.
• Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado.
• Escribe A o D en la primera columna O si no estás seguro(a), escribe NS (no estoy
seguro(a)).
PASO 1
A, D o NS
PASO 2
AoD
Enunciado
1. Los escalares tienen tanto magnitud como dirección.
2. Los vectores se pueden representar con segmentos de recta con
dirección.
3. El punto inicial de un vector es el punto donde comienza dicho
vector.
4. Los vectores se pueden usar para representar fuerzas aplicadas en
ángulo.
5. Los escalares son el resultado de la combinación de vectores.
6. Todo vector puede ser separado en un componente vertical y uno
8.
9.
10.
11.
12.
Paso 2
Después de que termines el Capítulo 8
• Relee cada enunciado y escribe A o D en la última columna.
• Compara la última columna con la primera. ¿Cambiaste de opinión sobre alguno de los
enunciados?
• En los casos en que hayas estado en desacuerdo con el enunciado, escribe en una hoja
aparte un ejemplo de por qué no estás de acuerdo.
Capítulo 8
4
Precálculo de Glencoe
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7.
horizontal.
Para sumar algebraicamente dos vectores, se suman todos los
números del primer vector y luego se suman todos los números
del segundo vector.
Algunas veces se usan razones trigonométricas para trabajar con
vectores.
El producto punto (o producto escalar) de dos vectores es un
escalar.
Los vectores permiten representar fuerzas en un espacio
tridimensional.
Para calcular el par de torsión se usa el producto punto
(o producto escalar).
El área de un paralelogramo se puede calcular utilizando el
producto punto.
NAME
DATE
8-1
PERIOD
Study Guide and Intervention
Introduction to Vectors
Geometric Vectors A vector is a quantity that has both magnitude and direction. The
magnitude of a vector is the length of a directed line segment, and the direction of a vector
is the directed angle between the positive x-axis and the vector. When adding or subtracting
vectors, you can use the parallelogram or triangle method to find the resultant.
a. v = 60 pounds of force at 125° to
the horizontal
b. w = 55 miles per hour at a bearing
of S45°E
Using a scale of 1 cm.: 20 mi/h, draw
and label a 55 ÷ 20 or 2.75-centimeter
arrow 45° east of south.
Using a scale of 1 cm: 20 lb, draw and
label a 60 ÷ 20 or 3-centimeter arrow
in standard position at a 125° angle to
the x-axis.
/
Z
1 cm: 20 mi/h
0
W
8
&
45°
X
125°
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1 cm: 20 lb
0 Y
4
Exercises
Use a ruler and a protractor to draw an arrow diagram for each quantity
described. Include a scale on each diagram.
1. r = 30 meters at a bearing of N45°W
2. t = 150 yards at 40° to the horizontal
Find the resultant of each pair of vectors using either the triangle or
parallelogram method. State the magnitude of the resultant in centimeters
and its direction relative to the horizontal.
3.
4.
F
B
G
C
Chapter 8
5
Glencoe Precalculus
Lesson 8-1
Example
Use a ruler and a protractor to draw an arrow diagram for each
quantity described. Include a scale on each diagram.
NAME
DATE
8-1
Study Guide and Intervention
PERIOD
(continued)
Introduction to Vectors
Vector Applications
Vectors can be resolved into horizontal and vertical components.
Example
Suppose Jamal pulls on the ends of a rope tied to a dinghy with a
force of 50 Newtons at an angle of 60° with the horizontal.
a. Draw a diagram that shows the resolution of the force Jamal exerts into its
rectangular components.
Jamal’s pull can be resolved into a horizontal
pull x forward and a vertical pull y upward
as shown.
50 N
y
60°
x
b. Find the magnitudes of the horizontal and vertical components of the force.
The horizontal and vertical components of the force form a right triangle.
Use the sine or cosine ratios to find the magnitude of each force.
⎪x⎥
cos 60° = −
50
Right triangle definitions of cosine and sine
⎪y⎥
sin 60° = −
50
⎪x⎥ = 50 cos 60°
Solve for x and y.
⎪y⎥ = 50 sin 60°
⎪x⎥ = 25
Use a calculator.
⎪y⎥ ≈ 43.3
Exercises
Draw a diagram that shows the resolution of each vector into its rectangular
components. Then find the magnitudes of the vector’s horizontal and vertical
components.
1. 7 inches at a bearing of 120°
from the horizontal
2. 2.5 centimeters per hour at a bearing of
N50°W
3. YARDWORK Nadia is pulling a tarp along level ground with a force of 25 pounds
directed along the tarp. If the tarp makes an angle of 50° with the ground, find the
horizontal and vertical components of the force. What is the magnitude and direction
of the resultant?
4. TRANSPORTATION A helicopter is moving 15° north of east with a velocity of 52 km/h.
If a 30-kilometer per hour wind is blowing from a bearing of 250°, find the helicopter’s
resulting velocity and direction.
Chapter 8
6
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The magnitude of the horizontal component is about 25 Newtons, and the
magnitude of the vertical component is about 43 Newtons.
NAME
8-1
DATE
PERIOD
Practice
Introduction to Vectors
Use a ruler and a protractor to draw an arrow diagram for each quantity
described. Include a scale on each diagram.
2. t = 100 pounds of force at 60° to the
horizontal
Lesson 8-1
1. r = 60 meters at a bearing of N45°E
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. GROCERY SHOPPING Caroline walks 45° north of west for 1000 feet and
then walks 200 feet due north to go grocery shopping. How far and at what
north of west quadrant bearing is Caroline from her apartment?
4. CONSTRUCTION Roland is pulling a crate of construction materials with a
force of 60 Newtons at an angle of 42° with the horizontal.
a. Draw a diagram that shows the resolution of the force Roland exerts into
its rectangular components.
b. Find the magnitudes of the horizontal and vertical components of the force.
5. AVIATION An airplane is flying with an airspeed of 500 miles per hour on a heading
due north. If a 50-mile per hour wind is blowing at a bearing of 270°, determine
the velocity and direction of the plane relative to the ground.
Chapter 8
7
Glencoe Precalculus
NAME
8-1
DATE
PERIOD
Word Problem Practice
Introduction to Vectors
1. SAILING A captain sails a boat east for
200 kilometers at a bearing of 150°. Use
a ruler and protractor to draw an arrow
diagram for the quantities described.
Include a scale on the diagram.
3. CANOEING A person in a canoe wants
to cross a 65-foot-wide river. He begins
to paddle straight across the river at
1.2 m/s while a current is flowing
perpendicular to the canoe. If the
resulting velocity of the canoe is 3.2 m/s,
what is the speed of the current to the
nearest tenth?
4. TRAVEL Karrie is pulling her luggage
across the airport floor. She applies a
22-newton force to the handle of the
bag when the bag makes a 72-degree
angle with the floor. What is the
magnitude of the force that moves the
luggage straight forward? What effect
would it have if Karrie moved the
handle closer to the floor, decreasing
the angle?
2. FARMING Two tractors are removing a
tree stump as shown. One tractor pulls
with a force of 2000 newtons, and the
other tractor pulls with a force of
1500 newtons. The angle between the
two tractors is 40°.
1500 N
20°
20°
a. What is the sum of the horizontal
components of the tractors? What is
the sum of the vertical components?
6. SKATEBOARDING Meredith is
skateboarding along a path 20° north
of east for 35 meters. She then changes
paths and travels for 45 meters along a
path 30° north of east.
b. What is the resulting force on the
tree stump?
a. Use a ruler and protractor to draw
an arrow diagram representing the
situation.
c. Would changing the angle of the
tractors affect the magnitude of the
resulting force if the angle between
the tractors remained 40°? Explain.
b. Find the resulting distance and the
direction of her path.
Chapter 8
8
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. ORIENTEERING In an orienteering
competition, Jada walks N70°W for
200 meters. She then walks due east for
90 meters. How far and at what bearing
is Jada from her starting point?
2000 N
NAME
DATE
8-1
PERIOD
Enrichment
More Than Two Forces Acting on an Object
Example
CONSTRUCTION Kendra is pulling on a box with a force
of 80 newtons at an angle of 70° with the ground at the same time that
Kyle is pulling on the box with a force of 100 newtons at an angle of 150°
with the ground. A third force of 120 N acts at an angle of 180°. Find
the magnitude and direction of the resultant force acting on the box.
80 N
100 N
120 N
180°
150°
70°
First, add two of the vectors. The order in which the vectors are added does
not matter.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Add the 80 N vector and the
100 N vector first.
Now add the resulting vector
to the 120 N vector.
100 N
a
r
a
80 N
120 N
The resultant force is 219 N at a direction of 145°.
Find the magnitude and direction of the resultant force
acting on each object.
1. DOGS Three dogs are pulling a wagon. One dog acts with 40 N at 50°
on the wagon. The second dog acts with 100 N at 110°. The third dog
acts with 10 N at 150°. Find the magnitude and direction of the
resultant force.
2. MOVING Three men are trying to move a sofa. One man is pushing
on the sofa with a force of 40 N at an angle of 50° with the ground.
A second man exerts a force of 100 N at 110°, and a third man exerts
a force of 10 N at 150°. Find the magnitude and direction of the
resultant force.
Chapter 8
9
Glencoe Precalculus
Lesson 8-1
Three or more forces may work on an object at one time. Each of these forces
can be represented by a vector. To find the resultant vector that acts upon the
object, you can add the individual vectors two at a time.
NAME
DATE
8-2
PERIOD
Study Guide and Intervention
Vectors in the Coordinate Plane
Vectors in the Corrdinate Plane
The magnitude of a vector in the
coordinate plane is found using the Distance Formula.
Example 1
⎯⎯⎯ with initial point X(2, −3) and
Find the magnitude of XY
terminal point Y(-4, 2).
⎯⎯⎯ using the Distance Formula.
Determine the magnitude of XY
⎪XY
⎯⎯⎯⎥ = √(x
- x1)2 + (y2 - y1)2
2
=
4
- 2) + [2 - (-3)]
√(-4
2
2
Y
y
2
(-6) 2 + 5 2
= √
x
0
−4 −2
or about 7.8 units
= √61
Example 2
4
−2
−4
⎯⎯⎯as an ordered pair.
Represent XY
⎯⎯⎯ = 〈x2 - x1, y2 - y1〉
XY
= 〈-4 - 2, 2 - (-3)〉
= 〈-6, 5〉
2
X
Component form
(x1, y1) = (2, −3) and (x2, y2) = (−4, 2)
Subtract.
Find each of the following for s = 〈4, 2〉 and t = 〈-1, 3〉.
s + t = 〈4, 2〉 + 〈-1, 3〉
= 〈4 + (-1), 2 + 3〉 or 〈3, 5〉
Substitute.
Vector addition
b. 3s + t
3s + t = 3〈4, 2〉 + 〈-1, 3〉
Substitute.
= 〈12, 6〉 + 〈-1, 3〉
Scalar multiplication
= 〈11, 9〉
Vector addition
Exercises
⎯⎯⎯ with the
Find the component form and magnitude of the vector AB
given initial and terminal points.
1. A(12, 41), B(52, 33)
2. A(-15, 0), B(7, -19)
Find each of the following for f = 〈4, -2〉, g = 〈24, 21〉, and h = 〈-1, -3〉.
3. f - g
4. 8g - 2f + 3h
5. 2g + h
6. f - 2(g + 2h)
Chapter 8
10
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a. s + t
NAME
DATE
8-2
Study Guide and Intervention
PERIOD
(continued)
Vectors in the Coordinate Plane
Unit Vectors A vector that has a magnitude of 1 unit is called a unit vector.
A unit vector in the direction of the positive x-axis is denoted as i = 〈1, 0〉, and
a unit vector in the direction of the positive y-axis is denoted as j = 〈0, 1〉.
Vectors can be written as linear combinations of unit vectors by first writing
the vector as an ordered pair and then writing it as a sum of the vectors i and j.
Example 1
Find a unit vector u with the same direction as v = 〈-4, -1〉.
1
u=−
v
Unit vector with the same direction as v
1
= −
〈-4, -1〉
⎪〈-4, -1〉⎥
Substitute.
1
= −
〈-4, -1〉
√
(-4)2 + (-1)2
a2 + b2
⎪〈a, b〉⎥ = √
1
=−
〈-4, -1〉
Simplify.
√
17
〈
-4 17 - 17
−, −〉
〈
〉
17
17
17
-1
-4
= −
,−
or
√
17 √
√
√
Scalar multiplication
Example 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
⎯⎯⎯ be the vector with initial point M(2, 2) and terminal
Let MP
⎯⎯⎯ as a linear combination of the vectors i and j.
point P(5, 4). Write MP
⎯⎯⎯⎯
.
First, find the component form of MP
⎯⎯⎯⎯
= 〈x2 - x1, y2 - y1〉
MP
Component form
= 〈5 - 2, 4 - 2〉 or 〈3, 2〉
(x1, y1) = (2, 2) and (x2, y2) = (5, 4)
Then rewrite the vector as a linear combination of the standard unit vectors.
⎯⎯⎯⎯
= 〈3, 2〉
MP
= 3i + 2j
Component form
〈a, b〉 = ai + bj
Exercises
Find a unit vector u with the same direction as the given vector.
1. p = 〈4, -3〉
2. w = 〈10, 25〉
⎯⎯⎯⎯ be the vector with the given initial and terminal points. Write MN
⎯⎯⎯⎯
Let MN
as a linear combination of the vectors i and j.
3. M(2, 8), N(-5, -3)
4. M(0, 6), N(18, 4)
Find the component form of v with the given magnitude and direction angle.
5. |v| = 18, θ = 240°
6. |v| = 5, θ = 95°
Find the direction angle of each vector to the nearest tenth.
7. -4i + 2j
Chapter 8
8. 〈2, 17〉
11
Glencoe Precalculus
Lesson 8-2
⎪ v⎥
NAME
8-2
DATE
PERIOD
Practice
Vectors in the Coordinate Plane
⎯⎯⎯ with the given initial and
Find the component form and magnitude of AB
terminal points.
1. A(2, 4), B(-1, 3)
2. A(4, -2), B(5, -5)
3. A(-3, -6), B(8, -1)
Find each of the following for v = 〈2, -1〉 and w = 〈-3, 5〉.
4. 3v
5. w - 2v
6. 4v + 3w
7. 5w - 3v
Find a unit vector u with the same direction as v.
8. v = 〈-3, 6〉
9. v = 〈-8, -2〉
10. D(4, -5), E(6, -7)
11. D(-4, 3), E(5, -2)
12. D(4, 6), E(-5, -2)
13. D(2, 1), E(3, 7)
Find the component form of v with the given magnitude and direction angle.
14. |v| = 12, θ = 42°
15. |v| = 8, θ = 132°
16. GARDENING Anne and Henry are lifting a stone statue and moving it to
a new location in their garden. Anne is pushing the statue with a force of
120 newtons at a 60° angle with the horizontal while Henry is pulling the
statue with a force of 180 newtons at a 40° angle with the horizontal.
What is the magnitude of the combined force they exert on the statue?
Chapter 8
12
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
⎯⎯⎯ be the vector with the given initial and terminal points. Write DE
⎯⎯⎯
Let DE
as a linear combination of the vectors i and j.
NAME
8-2
DATE
PERIOD
Word Problem Practice
Vectors in the Coordinate Plane
1. TRACK Monica is throwing the javelin
in a track meet. While running at
4 meters per second, she throws the
javelin with a velocity of 28 meters
per second at an angle of 48°.
4. AIRPLANES An airplane is traveling
300 kilometers per hour due east. A
wind is blowing 35 kilometers per hour
75° south of west. What is the resulting
speed of the airplane?
a. What is the resultant speed of
the javelin?
6. KAYAKING Walter is kayaking across
a river that has a current of 2.5 meters
per second. He is paddling at a rate of
4 meters per second perpendicular to
the shore.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. What is the resultant direction of
the javelin?
end
2.5 m/s
2. TRANSPORTATION Jordyn is riding
the bus to school. The bus travels
north for 4.5 miles, east for 2 miles,
and then 30° north of east for 1.5 miles.
Express Jordyn’s commute as a linear
combination of unit vectors i and j.
4 m/s
θ
start
a. What is the resultant velocity of
the kayak?
b. At what angle will Walter be moving
with respect to the shore?
3. HIKING Amel is hiking in the forest.
He hikes 2 miles west and then hikes
3.4 miles north. If he would have hiked
diagonally to reach the same ending
point, how much shorter would his hike
have been?
Chapter 8
c. If Walter wants to land directly in
front of his starting point, at what
angle with respect to the shore
should he kayak?
13
Glencoe Precalculus
Lesson 8-2
5. FLYING To reach a destination, a pilot
is plotting a course that will result in a
velocity of 450 miles per hour at an
angle of 30° north of west. The wind is
blowing 50 miles per hour to the north.
Find the direction and speed the pilot
should set to achieve the desired
resultant.
48°
NAME
DATE
8-2
PERIOD
Enrichment
Friction and the Normal Force
According to Newton’s first law of motion, if an object is moving with constant velocity, then
all the forces in the system are balanced. If the object is not moving at a constant velocity,
then the forces are unbalanced. So, if a person is pushing on an object and the object is
moving at a constant velocity, the force of the person pushing on the object is balanced
by the force of friction.
The coefficient of friction μ (pronounced “mu”) is the ratio of the force of friction between
two surfaces and the force pushing the surfaces together. When an object is resting on the
ground, the force pushing them together is found by multiplying the mass (in kilograms) by
acceleration due to gravity, -9.8 m/s2. The normal force n is the force of the ground
pushing back up on the object.
Example
Consider a person pulling a 7-kilogram wagon with
a constant velocity as shown in the diagram. The
person is pulling with a force of 25 newtons and
the wagon handle makes a 40° angle with the
horizontal. What is the coefficient of friction
for the situation?
25 N
40°
25 N
16.1 N
52.5 N
40°
-19.2 N
19.2 N
-68.6 N
x = 25 cos 40°
y = 25 sin 40°
≈ 19.2
x-component
≈ 16.1
y-component
Force due to gravity = mass × acceleration due to gravity
7(-9.8) = -68.6
n = 68.6 - 16.1 = 52.5
Normal force
19.2
μ = − about 0.37
Coefficient of friction
52.5
Therefore, the coefficient of friction is 0.37.
Exercises
1. What is the coefficient of friction for a person pulling a 41-kilogram child on a sled if the
angle of the rope is 45° and the force of the pull is 50 newtons?
2. What is the coefficient of friction for a person pushing a 25-kilogram box across a
carpeted floor with a force of 75 newtons if the force is being applied at 25°?
Chapter 8
14
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A force diagram can be used to represent this situation. Calculate the x-component and the
y-component of the 25-newton force. Because the wagon is moving at a constant velocity, the
force of friction has to be equal to, but opposite the x-component. The force of gravity is
balanced by the sum of the y-component and the normal force.
NAME
DATE
8-3
PERIOD
Study Guide and Intervention
Dot Products and Vector Projections
Dot Product
The dot product of a = 〈a 1, a 2〉 and b = 〈b 1, b 2〉 is defined as
a · b = a 1b 1 + a 2b 2. The vectors a and b are orthogonal if and only if a · b = 0.
a·b
If θ is the angle between nonzero vectors a and b, then cos θ = −
.
|a| |b|
Find the dot product of u and v. Then determine if u and v
Example 1
are orthogonal.
a. u = 〈5, 1〉, v = 〈-3, 15〉
u · v = 5(−3) + 1(15)
=0
Since u · v = 0, u and v are orthogonal.
Example 2
b. u = 〈4, 5〉, v = 〈8, -6〉
u · v = 4(8) + 5(-6)
=2
Since u · v ≠ 0, u and v are not orthogonal.
Find the angle θ between vectors u and v if u = 〈5, 1〉
and v = 〈−2, 3〉.
u·v
cos θ = −
Angle between two vectors
〈5, 1〉 · 〈-2, 3〉
|〈5, 1〉| |〈-2, 3〉|
cos θ = −
u = 〈5, 1〉 and v = 〈−2, 3〉
-10 + 3
cos θ = −
Evaluate.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
√
26 √
13
-10 + 3
√
26 √
13
θ = cos −1 − or about 112°
Simplify and solve for θ.
Lesson 8-3
|u| |v|
The measure of the angle between u and v is about 112°.
Exercises
Find the dot product of u and v. Then determine if u and v are orthogonal.
1. u = 〈2, 4〉, v = 〈−12, 6〉
2. u = -8i + 5j, v = 3i −6j
Use the dot product to find the magnitude of the given vector.
3. a = 〈9, 3〉
4. c = 〈−12, 4〉
Find the angle θ between u and v to the nearest tenth of a degree.
5. u = 〈-3, -5〉, v = 〈7, 12〉
Chapter 8
6. u = 13i - 5j, v = 6i + 2j
15
Glencoe Precalculus
NAME
DATE
8-3
PERIOD
Study Guide and Intervention
(continued)
Dot Products and Vector Projections
Vector Projection
A vector projection is the decomposition of a vector u into two
perpendicular parts, w1 and w2, in which one of the parts is parallel to another vector v.
When you find the projection of u onto v, you are finding a component of u that is parallel
to v. To find the projection of u onto v, use the formula:
u·v
proj v u = −
v.
2
( |v| )
Example
Find the projection of u = 〈8, 6〉 onto v = 〈2, −3〉. Then write u as
the sum of two orthogonal vectors, one of which is the projection of u onto v.
Step 1 Find the projection
of u onto v.
Step 2 Find u − proj v u.
〈
108 72
,−
= 〈−
13 13 〉
4 6
= 〈8, 6〉 − - −
,−
u·v
proj v u = −
v
|v| 2
)
(
〈8, 6〉 · 〈2, -3〉
13 13
〉
= −
〈2, -3〉
2
|〈2, -3〉|
〈
〉
2
4 6
〈2, -3〉 or - −
,−
= -−
13
〈
〉
13 13
〈
〉 〈 13
〉
108 72
4 6
4 6
,− .
, − and u = - −
,− + −
Therefore, proj v u is - −
13 13
13 13
13
Find the projection of u onto v. Then write u as the sum of two orthogonal
vectors, one of which is the projection of u onto v.
1. u = 〈3, 2〉, v = 〈-4, 1〉
2. u = 〈−7, 3〉, v = 〈8, 5〉
3. u = 〈1, 1〉, v = 〈9, -7〉
4. u = 7i - 9j, v = 12i + j
5. u = −8i + 2j, v = 6i + 13j
Chapter 8
16
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
8-3
DATE
PERIOD
Practice
Dot Products and Vector Projections
Find the dot product of u and v. Then determine if u and v are orthogonal.
1. u = 〈3, 6〉, v = 〈−4, 2〉
2. u = -i + 4j, v = 3i − 2j
3. u = 〈2, 0〉, v = 〈−1, −1〉
Find the angle θ between u and v to the nearest tenth of a degree.
4. u = 〈−1, 9〉, v = 〈3, 12〉
5. u = 〈−6, −2〉, v = 〈2, 12〉
6. u = 27i + 14j, v = i − 7j
7. u = 5i − 4j, v = 2i + j
8. u = 〈4, 8〉, v = 〈−1, 2〉
Lesson 8-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find the projection of u onto v. Then write u as the sum of two orthogonal
vectors, one of which is the projection of u onto v.
9. u = 〈62, 21〉, v = 〈−12, 4〉
10. u = 〈−2, −1〉, v = 〈−3, 4〉
11. TRANSPORTATION Train A and Train B depart from the same station. The path that
train A takes can be represented by 〈33, 12〉. If the path that train B takes can be
represented by 〈55, 4〉, find the angle between the pair of vectors.
12. PHYSICS Janna is using a force of 100 pounds to push a cart up a ramp. The
ramp is 6 feet long and is at a 30° angle with the horizontal. How much work
is Janna doing in the vertical direction? (Hint: Use the sine ratio and the formula
W = F · d.)
Chapter 8
17
Glencoe Precalculus
NAME
8-3
DATE
PERIOD
Word Problem Practice
Dot Products and Vector Projections
1. SUBMARINES The path of a submarine
can be described by the vector v = 〈8, 3〉.
If the submarine then changes direction
and travels along the vector u = 〈2, 5〉,
what is the distance traveled by the
submarine?
4. TRAVELING A pilot is carrying a bag
weighing 150 newtons up a flight of
stairs. The staircase covers a horizontal
distance of 8 meters and a vertical
distance 7.5 meters.
a. What is the length of the staircase?
2. TARGETS Two clay pigeons are thrown
at the same time. If the path of the clay
pigeons can be represented by the vectors
p = 〈42, 58〉 and c = 〈59, 73〉, what was
the measure of the angle between the clay
pigeons?
b. What is the measure of the angle
made by the staircase?
c. How much work is done by the pilot?
3. BOATING Shea is pulling a boat along
a dock using a rope. She exerts a force
of 200 newtons on the rope and pulls the
boat 10 meters.
a. Determine the amount of work done
if the angle of the rope is at 40° with
the horizontal; 90° with the horizontal.
60°
6. SCAVENGER HUNT During a
scavenger hunt, Alexis and Marty go in
different directions. If the path that
Alexis takes can be represented by
〈9, 18〉 and the path taken by Marty
can be represented by 〈-15, 12〉, who
travels the farthest distance?
b. Use your results from part a to explain
why as the angle increases the work
decreases.
Chapter 8
18
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. CARNIVALS A slide at a carnival has
an incline of 60°. A 50-pound girl gets
part way down the slide and stops.
Ignoring the force of friction, what is
the magnitude of the force that is
required to keep her from sliding
down farther?
NAME
DATE
8-3
PERIOD
Enrichment
Vector Equations
Let a, b, and c be fixed vectors. The equation f(x) = a - 2xb + x 2c
defines a vector function of x. For the values of x shown, the
assigned vectors are given below.
x
f(x)
-2
-1
0
1
2
a + 4b + 4c
a + 2b + c
a
a - 2b + c
a - 4b + 4c
If a = 〈0, 1〉, b = 〈1, 1〉, and c = 〈2, –2〉, the resulting vectors for the
values of x are as follows.
x
f(x)
-2
-1
0
1
2
〈12, -3〉
〈4, 1〉
〈0, 1〉
〈0, -3〉
〈4, -11〉
For each of the following, complete the table of resulting vectors.
1. f(x) = x 3a - 2x 2b + 3xc
a = 〈1, 1〉
b = 〈2, 3〉
x
–1
c = 〈3, –1〉
f(x)
0
1
2. f(x) = 2x 2a + 3xb - 5c
a = 〈0, 1, 1〉 b = 〈1, 0, 1〉
x
–2
c = 〈1, 1, 0〉
f(x)
Lesson 8-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
–1
0
1
3. f(x) = x 2c + 3xa - 4b
a = 〈1, 1, 1〉
b = 〈3, 2, 1〉
x
0
c = 〈0, 1, 2〉
f(x)
1
2
3
4. f(x) = x 3a - xb + 3c
a = 〈0, 1, –2〉
b = 〈1, –2, 0〉
c = 〈-2, 0, 1〉
x
0
f(x)
1
2
3
Chapter 8
19
Glencoe Precalculus
NAME
DATE
8-4
PERIOD
Study Guide and Intervention
Vectors in Three-Dimensional Space
Coordinates in Three Dimensions Ordered triples, like ordered pairs,
can be used to represent vectors. Operations on vectors represented by ordered
triples are similar to those on vectors represented by ordered pairs.
Example
HIKING The location of two hikers are represented by
the coordinates (10, 2, -5) and (7, -9, 3), where the coordinates are given
in kilometers.
a. How far apart are the hikers?
Use the Distance Formula for points in space.
AB =
(x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2
√
Distance Formula
= √
(7 - 10)2 + ((-9) - 2)2 + (3 - (-5))2
≈ 13.93
The hikers are about 14 kilometers apart.
(x1, y1, z1) = (10, 2, -5) and
(x2, y2, z2) = (7, -9, 3)
b. The hikers decided to meet at the midpoint between their paths.
What are the coordinates of the midpoint?
Use the Midpoint Formula for points in space.
x +x y +y z +z
10 + 7 2 + (-9) -5 + 3
, −, − ) = ( −, −, − )
(−
2
2
2
2
2
2
1
2
1
2
1
2
(x1, y1, z1) = (10, 2, -5) and
(x2, y2, z2) = (7, -9, 3)
≈ (8.5, -3.5, -1)
Exercises
Plot each point in a three-dimensional coordinate system.
1. (3, 2, 1)
2. (4, −2, -1)
z
0
z
y
0
y
x
x
Find the length and midpoint of the segment with the given endpoints.
3. (8, -3, 9), (2, 8, -4)
4. (-6, -12, -8), (7, -2, -11)
Chapter 8
20
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The midpoint is at the coordinates (8.5, -3.5, -1).
NAME
DATE
8-4
PERIOD
Study Guide and Intervention
(continued)
Vectors in Three-Dimensional Space
Vectors in Space Operations on vectors represented by ordered triples are similar to
those on vectors represented by ordered pairs. Three-dimensional vectors can be added,
subtracted, and multiplied by a scalar in the same ways. In space, a vector v in standard
position with a terminal point located at (v1, v2, v3) is denoted by 〈v1, v2, v3〉. Thus, the
zero vector is 0 = 〈0, 0, 0〉 and the standard unit vectors are i = 〈1, 0, 0〉, j = 〈0, 1, 0〉,
and k = 〈0, 0, 1〉. The component form of v can be expressed as a linear combination
of these unit vectors, 〈v1, v2, v3〉 = v1i + v2 j + v3k.
Example
⎯⎯⎯ with initial
Find the component form and magnitude of AB
point A(-3, 5, 1) and terminal point B(3, 2, -4). Then find a unit vector in the
⎯⎯⎯.
direction of AB
⎯⎯⎯⎯ = 〈x2 - x1, y2 - y1, z2 - z1〉
AB
Component form of vector
= 〈3 - (-3), 2 - 5, -4 - 1〉 or 〈6, -3, -5〉
(x1, y1, z1) = (-3, 5, 1) and (x2, y2, z2) = (3, 2, −4)
⎯⎯⎯⎯ is
Using the component form, the magnitude of AB
⎪AB
.
⎯⎯⎯⎯⎥ = √
62 + (-3)2 + (-5)2 or √70
⎯⎯⎯
AB = 〈6, -3, -5〉
Using this magnitude and component form, find a
⎯⎯⎯⎯.
unit vector u in the direction of AB
⎯⎯⎯⎯
AB
u=−
Unit vector in the direction of ⎯⎯⎯
AB
⎪AB
⎯⎯⎯⎯⎥
√
70
3 70
3 70
70
, - −, - − 〉
〈−
35
70
14
√
√
√
⎯⎯⎯
AB = 〈6, -3, -5〉 and ⎪ ⎯⎯⎯
AB ⎥ = √
70
Exercises
⎯⎯⎯ with the given initial
Find the component form and magnitude of AB
⎯⎯⎯.
and terminal points. Then find a unit vector in the direction of AB
1. A(-10, 3, 9), B(8, -7, 3)
2. A(-1, -4, -7), B(8, 4, 10)
Lesson 8-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
〈6, -3, -5〉
= − or
Find each of the following for x = 3i + 2j - 5k, y = i - 5j + 7k,
and z = -2i + 12j + 4k.
3. 3x + 2y - 4z
Chapter 8
4. -6y + 2z
21
Glencoe Precalculus
NAME
DATE
8-4
PERIOD
Practice
Vectors in Three-Dimensional Space
Plot each point in a three-dimensional coordinate system.
1. (-3, 4, -1)
2. (2, 0, -5)
z
0
y
z
y
0
x
x
Locate and graph each vector in space.
3. 〈4, 7, 6〉
4. 〈4, -2, 6〉
z
z
y
y
x
x
5. A(2, 1, 3), B(-4, 5, 7)
6. A(4, 0, 6), B(7, 1, -3)
7. A(-4, 5, 8), B(7, 2, -9)
8. A(6, 8, -5), B(7, -3, 12)
Find the length and midpoint of the segment with the given endpoints.
9. (3, 4, -9), (-4, 7, 1)
10. (-17, -3, 2), (3, -9, 5)
Find each of the following for v = 〈2, -4, 5〉 and w = 〈6, -8, 9〉.
11. v + w
12. 5v - 2w
13. PHYSICS Suppose that the force acting on an object can be expressed by the vector
〈85, 35, 110〉, where each measure in the ordered triple represents the force in pounds.
What is the magnitude of this force?
Chapter 8
22
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
⎯⎯⎯⎯ with the given initial
Find the component form and magnitude of AB
⎯⎯⎯⎯.
and terminal points. Then find a unit vector in the direction of AB
NAME
DATE
8-4
PERIOD
Word Problem Practice
Vectors in Three-Dimensional Space
1. TRAVELING A family from Des Moines,
Iowa, is driving to Tampa, Florida.
According to the car’s GPS, Des Moines
is at (93.65˚, 41.53˚, 955 ft) and Tampa
is at (82.53˚, 27.97˚, 19.7 ft). Determine
the longitude, latitude, and altitude of
the halfway point.
3. AIRPLANE Safety regulations require
airplanes to be at least a half mile
apart when they are in the air. Two
airplanes near an airport can be
represented by the points (300, 455,
2800) and (-250, 400, 5000), where
the coordinates are given in feet.
a. How far apart are the planes?
2. FARMING A farmer is using a bale
elevator to move bales of hay into the loft
of his barn. The opening of the loft door
is 18 feet away from where the bales will
be loaded onto the bale elevator, 3 feet to
the right of where the bales will be
loaded, and 24 feet above the ground.
The opening can be represented by the
point (18, 3, 24). The bales will be loaded
onto the elevator 3 feet above the ground.
This can be represented by (0, 0, 3).
b. Are they in violation of the
regulation?
4. ROBOTICS An underwater robot is
being used to explore parts of the ocean
floor. The robot is diving due north at
3 m/s at an angle of 65˚ with the surface
of the water. If the current is flowing at
5 m/s at an angle of 20˚ north of west,
what is the vector that represents the
resultant velocity of the underwater
robot? Let i point east, j point north,
and k point up.
24 ft
3 ft
3 ft
18 ft
5. ZIP-LINES A resort in Colorado has a
series of zip-lines that tourists can take
to travel through some wooded areas.
The platform of the first zip-line is
represented by the point (1.5, 0.5, 0.4)
and a second platform can be
represented by the point (1.8, 1, 0.2).
How long is the zip-line if the
coordinates are in miles?
a. To the nearest foot, how long should
the bale elevator be in order to reach
the opening?
b. If the bale elevator needs to be 2 feet
past the opening, to the nearest foot,
how long does the bale elevator have
to be?
c. If the bale elevator is only 27 feet
long and the only thing that can
be changed is the 18 feet that the
farmer is away from the opening, to
the nearest foot, how close does he
need to be in order to still have two
feet past the opening?
Chapter 8
6. BIKING A youth group is hosting a
team bike race with pairs of
competitors. Each team will switch
riders half-way through the race. If the
starting point can be represented
by (0, 0, 3) and the ending point can be
represented by the point (2, -1, 9), at
what point will the cyclists trade?
23
Glencoe Precalculus
Lesson 8-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2 ft
NAME
8-4
DATE
PERIOD
Enrichment
Basis Vectors in Three-Dimensional Space
The expression v = ru + sw + tz, is the sum of three vectors each multiplied by
a scalar, and is called a linear combination of the vectors u, w, and z.
Every vector v ∈ v3 can be written as a linear combination of any three nonparallel
vectors. The three nonparallel vectors, which must be linearly independent, are said
to form a basis for v3, which contains all vectors having 1 column and 3 rows.
Example
Write the vector v = 〈-1, -4, 3〉 as a linear combination of the
vectors u = 〈1, 3, 1〉, w = 〈1, -2, 1〉, and z = 〈-1, -1, 1〉.
r+s-t
〈-1, -4, 3〉 = r 〈1, 3, 1〉 + s 〈1, -2, 1〉 + t 〈-1, -1, 1〉 = 3r - 2s - t
r+s+t
(
)
-1 = r + s - t
-4 = 3r - 2s - t
3=r+s+t
Solving the system of equations yields the solution r = 0, s = 1, and t = 2.
So, v = w + 2z.
1. v = 〈-6, -2, 2〉, u = 〈1, 1, 0〉, w = 〈1, 0, 1〉, and z = 〈0, 1, 1〉
2. v = 〈5, -2, 0〉, u = 〈1, -2, 3〉, w = 〈-1, 0, 1〉, and z = 〈4, 2, -1〉
3. v = 〈1, -1, 2〉, u = 〈1, 2, -1〉, w = 〈2, 2, 1〉, and z = 〈1, 0, 1〉
Chapter 8
24
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Write each vector as a linear combination of the vectors u, w, and z.
NAME
8-4
DATE
PERIOD
Graphing Calculator Activity
Vector Transformations with Matrices
Vectors can be used to translate figures in space, and matrix multiplication can be used to
transform figures in space. A vertex matrix is a matrix whose columns are the coordinates of
the vertices of the figure with the x-coordinate represented by the first row, the y-coordinate
represented by the second row, and the z-coordinate represented by the third row.
Consider the pyramid shown at the right.
z
Use the coordinates of the vertices of the
pyramid to create a vertex matrix.
&
A(-2, -2, -2)
B(2, -2, -2)
0
"
% y
C(2, 2, -2)
#
$
D(-2, 2, -2)
x
E(0, 0, 2)
A
B
C
D E
x ⎡-2 2 2 -2 0⎤
⎢
The vertex matrix for the pyramid is y -2 -2 2 2 0 .
⎢
z ⎣-2 -2 -2 -2 2⎦
To reflect the image over the xz-plane, use the
transformation matrix B.
⎡ 1 0 0⎤
⎢
B = 0 -1 0
⎢
⎣0 0 1⎦
Enter B into a graphing calculator.
To find the reflected image, find BA. Verify that your
answer is correct by graphing the coordinates.
Exercises
1. Find the reflected pyramid above when
you use the transformation matrix B
below. Describe the result.
⎡1 0 0⎤
⎢
B= 0 1 0
⎢
⎣0 0 -1⎦
Chapter 8
2. Find the transformation matrix to reflect
over the yz-plane. Check your answer by
applying it to the pyramid above.
25
Glencoe Precalculus
Lesson 8-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Enter the vertex matrix into a graphing calculator.
NAME
DATE
8-5
PERIOD
Study Guide and Intervention
Dot and Cross Products of Vectors in Space
Dot Products in Space The dot product of two vectors in space is an extension
of the dot product of two vectors in a plane. Similarly, the dot product of two vectors
is a scalar. The dot product of
a = 〈a1, a2, a3〉 and b = 〈b1, b2, b3〉 is defined as a · b = a1b1 + a2b2 + a3b3.
The vectors a and b are orthogonal if and only if a · b = 0.
As with vectors in a plane, if θ is the angle between nonzero vectors a and b, then
a·b
cos θ = −
.
⎪a⎥ ⎪b⎥
Example 1
Find the dot product of u and v. Then determine if u and v
are orthogonal.
b. u = 〈3, −2, 1〉, v = 〈4, 5, −1〉
a. u = 〈-3, 1, 0〉, v = 〈2, 6, 4〉
u · v = u1v1 + u2v2 + u3v3
u · v = u1v1 + u2v2 + u3v3
= 3(4) + (−2)(5) + 1(−1)
= −3(2) + 1(6) + 0(4)
= 12 + (−10) -1 or 1
= −6 + 6 + 0 or 0
Since u · v ≠ 0, u and v are not orthogonal.
Since u · v = 0, u and v are orthogonal.
Example 2
Find the angle θ between vectors u and v if u = 〈4, 8, -3〉
and v = 〈9, −3, 0〉.
u·v
cos θ = −
Angle between two vectors
⎪u⎥ ⎪v⎥
u = 〈4, 8, -3〉 and v = 〈9, -3, 0〉
12
cos θ = −
Evaluate the dot product and magnitude.
√
89 √
90
12
θ = cos -1 −
or about 82.3°
89.5
Simplify and solve for θ.
The measure of the angle between u and v is about 82.3°.
Exercises
Find the dot product of u and v. Then determine if u and v are orthogonal.
1. u = 〈3, -2, 9〉, v = 〈1, 2, 4〉
2. u = 〈-2, -4, -6〉, v = 〈-3, 7, -4〉
3. u = 〈4, -3, 8〉, v = 〈2, -2, -3〉
4. u = 3i + 6j - 3k, v = -5i - 2j - 9k
Find the angle θ between vectors u and v to the nearest tenth of a degree.
5. u = 〈5, -22, 9〉, v = 〈14, 2, 4〉
6. u = 〈4, -5, 7〉, v = 〈11, -8, 2〉
7. u = -4i + 5j - 3k, v = -8i - 12j - 9k
8. u = i + 2j - k, v = -i + 4j - 3k
Chapter 8
26
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
〈4, 8, -3〉·〈9, -3, 0〉
cos θ = −−
⎪〈4, 8, -3〉⎥ ⎪〈9, -3, 0〉⎥
NAME
DATE
8-5
PERIOD
Study Guide and Intervention
(continued)
Dot and Cross Products of Vectors in Space
Cross Products Unlike the dot product, the cross product of two vectors
is a vector. This vector does not lie in the plane of the given vectors but is
perpendicular to the plane containing the two vectors.
Cross Product of Vectors in Space
If a = a1i + a2j + a3k and b = b1i + b2j + b3k, the cross product of a and b is the vector
a × b = (a2b3 - a3b2)i - (a1b3 - a3b1)j + (a1b2 - a2b1)k.
If two vectors have the same initial point and form the sides of a parallelogram, the
magnitude of the cross product will give you the area of the parallelogram.
If three vectors have the same initial point and form adjacent edges of a parallelepiped,
then the absolute value of the triple scalar product gives the volume. To find the triple
scalar product, use the same matrix set up that is used for cross products, but i, j, and
k are replaced by the third vector.
Example
Find the cross product of u = 〈0, 4, 1〉and v = 〈0, 1, 3〉.
Then show that u × v is orthogonal to both u and v.
i j k
u×v= 0 4 1
u = 0i + 4j + k and v = 0i + j + 3k
0 1 3
0 4
0 1
4 1
=
ij+
k
Determinant of a 3 × 3 matrix
1 3
0 1
0 3
= (12 - 1)i − (0 - 0) j + (0 − 0)k
Determinants of 2 × 2 matrices
⎪ ⎥
= 11i - 0j + 0k
Simplify.
= 11i or 〈11, 0, 0〉
Component form
To show that u × v is orthogonal to both u and v, find the dot
product of u × v with u and u × v with v.
(u × v) · u
(u × v) · v
= 〈11, 0, 0〉 · 〈0, 4, 1〉
= 〈11, 0, 0〉 · 〈0, 1, 3〉
= 11(0) + 0(4) + 0(1)
= 11(0) + 0(1) + 0(3)
=0+0+0
=0+0+0
=0
=0
Because both dot products are zero, the vectors are orthogonal.
Exercises
Find the cross product of u and v. Then show that u × v is orthogonal
to both u and v.
1. u = 〈2, 3, -1〉, v = 〈6, -2, -4〉
Lesson 8-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
⎪ ⎥ ⎪ ⎥ ⎪ ⎥
2. u = 〈5, 2, 8〉, v = 〈-1, 2, 4〉
Chapter 8
27
Glencoe Precalculus
NAME
8-5
DATE
PERIOD
Practice
Dot and Cross Products of Vectors in Space
Find the dot product of u and v. Then determine if u and v are orthogonal.
1. 〈-2, 0, 1〉 · 〈3, 2, -3〉
2. 〈-4, -1, 1〉 · 〈1, -3, 4〉
3. 〈0, 0, 1〉 · 〈1, -2, 0〉
Find the angle θ between vectors u and v to the nearest tenth of a degree.
4. u = 〈1, -2, 1〉,
v = 〈0, 3, -2〉
5. u = 〈3, -2, 1〉,
v = 〈-4, -2, 5〉
6. u = 〈2, -4, 4〉,
v = 〈-2, -1, 6〉
Find the cross product of u and v. Then show that u × v is orthogonal
to both u and v.
7. 〈1, 3, 4〉 × 〈-1, 0, -1〉
9. 〈3, 1, 2〉 × 〈2, -3, 1〉
8. 〈3, 1, -6〉 × 〈-2, 4, 3〉
10. 〈4, -1, 0〉 × 〈5, -3, -1〉
11. u = 〈9, 4, 2〉, v = 〈6, -4, 2〉
12. u = 〈2, 0, -8〉, v = 〈-3, -8, -5〉
13. Find the volume of the parallelepiped with adjacent edges represented by the
vectors 〈3, -2, 9〉, 〈6, -2, -7〉, and 〈-8, -5, -2〉.
14. TOOLS A mechanic applies a force of 35 newtons straight down to a ratchet
that is 0.25 meter long. What is the magnitude of the torque when the handle
makes a 20° angle above the horizontal?
Chapter 8
28
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find the area of the parallelogram with adjacent sides u and v.
NAME
8-5
DATE
PERIOD
Word Problem Practice
Dot and Cross Products of Vectors in Space
1. MECHANIC A mechanic is setting the
timing of an engine. He is using a ratchet
to turn the crankshaft. The ratchet is
0.5 meter long and the mechanic applies
22 newtons of force straight down on the
handle when the handle is at a 25° angle
with the horizontal. What is the
magnitude of the torque?
5. BICYCLING A cyclist applies a force
straight down on a bicycle pedal, as
shown in the diagram. The length to
the pedal’s axle is 0.2 meter and the
angle created with the vertical is 60°.
The magnitude of the torque is 150
newton meters. Find the force applied to
the pedal.
60° 9
3. MIRROR Two adjacent edges of a mirror
in a dressing room are represented by
the vectors 〈3, 4, 2〉 and 〈-4, 4, 3〉. What
is the area of the mirror?
6. ROCKETS Two rockets are launched
simultaneously. The first rocket starts
at the point (0, 1, 0) and after 1 second
is at the point (3, 7, 12). The second
rocket starts at the point (0, -1, 0)
and after 1 second is at the
point (3, -8, 10).
4. SCULPTURE A parallelepiped sculpture
is being created. When the sculpture is
set, three adjacent edges can be
represented by the vectors
t = 〈15, 12, 10〉, u = 〈13, -8, -5〉, and
a. What vector represents the path of
the first rocket?
v = 〈-9, 13, 12〉. What is the surface
area of the sculpture?
b. What vector represents the path of
the second rocket?
c. What is the measure of the angle
between the two rockets?
d. If the velocity of the rockets
remains constant, what vectors
would represent the rockets at
3 seconds?
Chapter 8
29
Glencoe Precalculus
Lesson 8-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. REVOLVING DOOR Erica is standing
in a revolving door that is not moving.
If Erica wants to produce just enough
torque to make the door rotate but wants
to apply the least amount of force, where
should she push on the door with respect
to the axis of rotation?
NAME
8-5
DATE
PERIOD
Enrichment
Linearly Dependent Vectors
The zero vector is 〈0, 0〉 in two dimensions and 〈0, 0, 0〉 in three dimensions.
A set of vectors is called linearly dependent if and only if there exist scalars,
not all zero, such that a linear combination of the vectors yields a zero vector.
Example
Are the vectors 〈-1, 2, 1〉, 〈1, -1, 2〉, and 〈0, -2, -6〉
linearly dependent?
Solve a〈-1, 2, 1〉 + b〈1, -1, 2〉 + c〈0, -2, -6〉 = 〈0, 0, 0〉.
-a + b
=0
2a - b - 2c = 0
a + 2b - 6c = 0
The above system does not have a unique solution. Any solution must satisfy
the conditions that a = b = 2c.
1
Hence, one solution is a = 1, b = 1, and c = −
.
2
1
〈-1, 2, 1〉 + 〈1, -1, 2〉 + −
〈0, -2, -6〉 = 〈0, 0, 0〉, so the three vectors are linearly
dependent.
2
1. 〈-2, 6〉, 〈1, -3〉
2. 〈3, 6〉, 〈2, 4〉
3. 〈1, 1, 1〉, 〈-1, 0, 1〉, 〈1, -1, -1〉
4. 〈1, 1, 1〉, 〈-1, 0, 1〉, 〈-3, -2, -1〉
5. 〈2, -4, 6〉, 〈3, -1, 2〉, 〈-6, 8, 10〉
9
6. 〈1, -2, 0〉, 〈2, 0, 3〉, -1, 1, −
Chapter 8
〈
30
4
〉
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Determine whether the given vectors are linearly dependent. Write yes or
no. If yes, give a linear combination that yields a zero vector.
NAME
DATE
8-5
PERIOD
Graphing Calculator Activity
Use Matrices to Find the Volume of Parallelepipeds
The volume of a parallelepiped with adjacent sides t = 〈t1, t2, t3〉, u = 〈u1, u2, u3〉,
and v = 〈v1, v2, v3〉 can be calculated by finding the determinant of the
matrix below.
⎡t1 t2 t3 ⎤
⎢u u u ⎢ 1 2 3
⎣v1 v2 v3 ⎦
Example
Find the volume of a parallelepiped with adjacent sides
t = 〈3, -2, 8〉, u = 〈4, 9, -1〉, and v = 〈-1, -5, -7〉.
Enter the data into the graphing calculator under matrix A.
2nd
2
[MATRIX]
ENTER
(–)
7
8
ENTER
ENTER
2nd
ENTER
4
ENTER
9
3
ENTER
ENTER
(–)
1
3
ENTER
ENTER
3
(–)
ENTER
1
ENTER
(–)
(–)
5
ENTER
[QUIT]
Now use the MATH menu under MATRIX to calculate the following determinant.
[MATRIX]
ENTER
2nd
[MATRIX]
ENTER
)
ENTER
The determinant is -350. Volume cannot be negative, so the volume is
350 cubic units.
Exercises
1. Find the volume of the parallelepiped with adjacent sides t = 〈6, -12, -9〉,
u = 〈4, 3, 9〉, and v = 〈2, 3, 1〉.
2. Find the volume of the parallelepiped with adjacent sides t = 〈8, -22, 90〉,
u = 〈-31, 3, 22〉, and v = 〈-65, 31, 0〉.
3. The volume of a parallelepiped is 112 cubic units. Three adjacent sides are
t = 〈-10, 3, 4〉, u = 〈-8, 7, 3〉, and v = 〈-6, -2, x〉. Is 3, 5, or 9 the correct
value of x?
Chapter 8
31
Glencoe Precalculus
Lesson 8-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2nd
NAME
DATE
8
PERIOD
Chapter 8 Quiz 1
SCORE
(Lessons 8-1 and 8-2)
1. The vector v has a magnitude of 13 millimeters and a
direction of 84°. Find the magnitude of its vertical and
horizontal components.
1.
2. Find a unit vector with the same direction as v = 〈6, -3〉.
A
〈−, - −〉 B 〈- −, −〉 C 〈- −, - −〉 D 〈−, - −〉
2 √5
5
√5
2 √5
5
5
√5
5
2 √5
5
√5
5
2
5
1
5
2.
3. 2a - b
3.
B
C
4.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4. −a + 2b
⎯⎯⎯⎯ with initial point
5. Find the component form of AB
A(1, -3) and terminal point B(-6, -8).
5.
⎯⎯⎯⎯ as a linear combination of the standard unit vectors
6. Write CD
for C(7, -4) and D(-8, 1).
6.
7. Two people are carrying a box. One person exerts a force
of 140 pounds at an angle of 65.5° with the horizontal.
The other person exerts a force of 115 pounds at an angle
of 58.3° with the horizontal. Find the net weight of the box.
8. Find each of the following for v = 〈6, -6〉 and w = 〈3, -4〉.
b. 2v + 3w
a. -5w
c.
NAME
8
4w - v
7.
8a.
8b.
8c.
DATE
PERIOD
Chapter 8 Quiz 2
SCORE
(Lesson 8-3)
1. Maggie is pulling a tarp along level ground with a force of
25 newtons. If the tarp makes an angle of 50° with the ground,
what are the vertical and horizontal components of the force?
Let a = 〈3, 8〉 and b = 〈-4, 6〉.
2. Which is the dot product of a and b?
A 36
B 0
C 〈-1, 14〉
D 〈7, 2〉
1.
2.
3. Find the angle θ between a and b to the nearest tenth.
3.
4. Find the projection of a onto b.
4.
5. SLEDDING A person is pulling a sled with a constant force of
35 newtons. The angle the rope makes with the sled is 40°,
and the sled is pulled 20 meters. Find the work done in joules.
5.
Chapter 8
Assessment
Find the resultant of each pair of vectors using either the
triangle or the parallelogram method. State the magnitude
in centimeters and its direction relative to the horizontal.
33
Glencoe Precalculus
NAME
DATE
8
PERIOD
Chapter 8 Quiz 3
SCORE
(Lesson 8-4)
1. Find the component form and magnitude of a vector with
initial point A(3, 4, 10) and terminal point B(8, 4, -2).
1.
2. Find 3x + y - z for x = 〈-1, 5, 2〉, y = 〈2, -3, 4〉,
and z = 〈-4, 1, 0〉.
2.
1
3. Find 2v + −
w - z for v = 2i - j + 5k, w = -3i + 4j - 6k
3
and z = 3j - 2k.
3.
4. Which is the midpoint between points (3, 9, -2) and (-4, 9, -6)?
A
(−72 , 0, 2)
(
1
, 9, -4
B (-12, 81, 12) C (-1, 18, -8) D - −
2
)
4.
5. Find the distance between points A(12, -9, 15) and B(1, 6, 2).
5.
NAME
PERIOD
Chapter 8 Quiz 4
SCORE
(Lesson 8-5)
1. Find the dot product of a and b if a = 〈7, -3, 8〉 and
b = 〈5, -2, -4〉. Then determine if a and b are orthogonal.
1.
2. Find the cross product of c and d if c = 〈5, -5, 4〉 and
d = 〈2, 3, -6〉. Verify that the resulting vector is
orthogonal to c and d.
2.
3. Find the measure of the angle between the vectors 〈6, -3, 1〉
and 〈8, 9, -11〉.
3.
4. A 32-newton force is applied straight down on the end of a
wrench that is 0.3 meters long. What is the magnitude of the
torque when the wrench is at a 35° angle with the horizontal?
4.
5. Find the area of a parallelogram with adjacent sides
〈5, 2, 8〉 and 〈-3, 4, 6〉.
A 8.5 units2
Chapter 8
B 17.4 units2
C 35.7 units2 D 63.2 units2
34
5.
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
8
DATE
NAME
8
DATE
PERIOD
Chapter 8 Mid-Chapter Test
SCORE
Part I Write the letter for the correct answer in the blank at the right of each question.
1. SLEDDING Jordyn pulls a sled with a force of 120 newtons at an angle
of 25° with the horizontal. Find the magnitude of the horizontal
component of the force.
B 56.0 N
C 88.3 N
D 108.8 N
1.
⎯⎯⎯⎯ be a vector with the given initial point A(8, −4) and terminal point
2. Let AB
⎯⎯⎯⎯ as a linear combination of the vectors i and j.
B(−2, −3). Write AB
J −6i + 7j
2.
3. Find the component form and magnitude of the vector with initial point
A(−6, 4) and terminal point B(−2, −1).
A 〈4, -5〉; √41
B 〈4, -5〉; 9
C 〈-4, 5〉; √
41
D 〈-4, 5〉; 9
3.
F 10i - j
G 6i - 7j
H −10i + j
4. Find the dot product of u = 〈8, 7〉 and v = 〈-3, -2〉. Then determine if u
and v are orthogonal.
F −9, orthogonal
G −9, not orthogonal
H −38, not orthogonal
J −38, orthogonal
4.
5. If u = 〈-8, 7〉 and v = 〈4, -6〉, find 2u - v.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A 〈−20, 20〉
B 〈20, −20〉
C 〈−12, 8〉
D 〈12, −8〉
5.
Part II
6. Find the projection of u = 〈7, −3〉 onto v = 〈4, 3〉. Then write
u as the sum of two orthogonal vectors, one of which is the
projection of u onto v.
6.
7. Find the measure of the angle θ between u = 〈9, 9〉 and v = 〈-7, 8〉
to the nearest tenth of a degree.
7.
8. Find the resultant of the pair of vectors using either the triangle
or parallelogram method.
8.
9. FOOTBALL With time running out in a game, Rodney runs with the
football at a speed of 3.8 meters per second and throws the ball at a
speed of 7 meters per second at an angle of 28° to the horizontal.
a. Write the component form of the vectors representing Rodney’s
velocity and the path of the ball.
9a.
b. What is the resultant speed and direction of the ball?
9b.
Chapter 8
35
Glencoe Precalculus
Assessment
A 50.7 N
NAME
DATE
PERIOD
8
Chapter 8 Vocabulary Test
SCORE
component form
opposite vectors
terminal point
components
ordered triple
triple scalar
cross product
orthogonal
true bearing
direction
parallel vectors
unit vector
dot product
quadrant bearing
vector
initial point
resultant
z-axis
magnitude
standard position
Choose the best term from the vocabulary list above to complete each sentence.
1.
1. When a vector is in its x and y components, the vector is
said to be in
.
2. The
2.
is the point where the vector begins.
3. Two vectors are
equal to zero.
3.
if their dot product is
4.
4. The length of a vector is known as its
.
5.
.
6. A point in space can be represented by a(n)
7. A(n)
8.
6.
.
7.
is a vector that has a length of one unit.
are vectors that have the same magnitude
and opposite directions.
9. The ending point of a vector is the
10. A vector is in
is at the origin.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5. Vectors that have the same or opposite direction but not
necessarily the same magnitude are known as
8.
9.
.
if its initial point
10.
Define each term in your own words.
11. true bearing
12. resultant
Chapter 8
36
Glencoe Precalculus
NAME
DATE
8
PERIOD
Chapter 8 Test, Form 1
SCORE
Write the letter for the correct answer in the blank at the right of each question.
1. Find the vertical component of v with a magnitude of 5 inches and a
direction angle of 32°.
A 2.65 in.
B 2.79 in.
C 4.24 in.
D 31.88 in.
1.
F
〈- −15 , −15 〉
G 〈8, -6〉
H
〈- −35 , −45 〉
J
〈−35 , - −45 〉
2.
3. Find the resultant of the pair of vectors using either the triangle
method or the parallelogram method. State the magnitude of the
resultant in centimeters.
A 2 cm
C 4.95 cm
B 3.5 cm
D 5.25 cm
a
3.
b
4. Find the measure of the angle θ between vectors a = 〈4, 6〉 and b = 〈2, 8〉
to the nearest tenth of a degree.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
F 19.7°
G 43.3°
H 70.4°
J 102.3°
4.
⎯⎯⎯⎯ with initial point A(1, 2)
5. Find the component form and magnitude of AB
and terminal point B(0, 3).
A 〈-1, 1〉; 1.41
B 〈1, -1〉; 2
C 〈-1, -1〉; 1.41
D 〈1,1〉; 2
5.
6. A force F1 of 9 newtons pulls due north. A force F2 of 12 newtons pulls
due east. Find the magnitude and direction of the resultant force.
F 15 N; 36.9°
G 15 N; 53.1°
H 21 N; 36.9°
J 21 N; 53.1°
6.
For Questions 7 and 8, find each of the following for v = 〈3, −4〉,
w = 〈3, −1〉, r = 〈2, 7, −2〉, and s = 〈-3, 4, 9〉.
7. 2v + w
A 〈6, −5〉
B 〈6, −6〉
C 〈9, −9〉
D 〈9, -10〉
7.
G 〈1, −13, −7〉
H 〈−5, −3, 11〉
J 〈5, 3, −11〉
8.
8. r - s
F 〈−1, 13, 7〉
9. Erin pushes a box up a ramp with a constant force of 60 newtons at a
constant angle of 25°. Find the work done in joules to move the box 5 meters.
A 126.7 J
B 139.9 J
C 225.8 J
D 271.9 J
9.
10. Find the cross product of v = 〈-1, 2, 4〉 and w = 〈-3, -1, 5〉.
F 〈14, -7, -5〉
Chapter 8
G 〈14, 7, 7〉
H 〈14, -7, 7〉
37
J 〈6, -7, 7〉
10.
Glencoe Precalculus
Assessment
2. Find a unit vector u with the same direction as v = 〈−3, 4〉.
NAME
DATE
8
Chapter 8 Test, Form 1
PERIOD
(continued)
11. Find the measure of the angle θ between u = 〈2, 1, 3〉 and v = 〈−4, 3, 0〉
to the nearest tenth of a degree.
A -15.5°
B 36.1°
C 54.0°
D 105.5°
11.
For Questions 12 and 13, find each dot product. Then determine if
the vectors are orthogonal.
12. 〈2, 3〉·〈4, 5〉
F 22, orthogonal
G 22, not orthogonal
H 23, orthogonal
J 23, not orthogonal
12.
13. 〈3, 0, −2〉 · 〈4, −2, 6〉
A 0, orthogonal
B 0, not orthogonal
C 9, orthogonal
D 9, not orthogonal
13.
14. A constant force of 42 newtons is being applied on an object in
the direction of due east at the same time that a constant force
of 35 newtons is being applied on the object in the direction of
due north. What is the magnitude and direction of the force?
F 38.5 N, 39.8°
G 38.5 N, 50.2°
H 54.7 N, 39.8°
J 54.7 N, 50.2° 14.
15.
16. A cruise ship’s path is represented by the vector 〈9, 17〉.
It then follows a new path represented by the vector
〈12, 8〉. What is the resultant path?
F 〈3, 9〉
G 〈21, 25〉
H 〈−3, 9〉
J 〈−21, 25〉
16.
16
24
, −−
〈−
13 〉
13
17.
17. Find the projection of u = 〈4, 2〉 onto v = 〈−3, 2〉.
A
3 2
,−
〈− −
13 13 〉
B
4 2
,−
〈−
13 13 〉
C
32
16
, −−
〈− −
13
13 〉
18. Find the area of the parallelogram with adjacent sides
u = 〈-3, 4, 8〉 and v = 〈9, -1, 6〉.
F 32 units2
G 76 units2
H 82.7 units2
D
J 101.1 units2
18.
19. The position of one airplane is represented by 〈9, 5, 3〉 and a second
airplane is represented by 〈−7, 7, 4〉. Determine the distance between
the planes if one unit represents one mile.
A 9.5 mi
B 14.0 mi
C 15.8 mi
D 16.2 mi
19.
20. Find the volume of the parallelepiped with adjacent edges
u = 〈2, 3, 0〉, v = 〈−4, 5, 1〉, and w = 〈−2, 3, 4〉.
F 8 units3
G 20 units3
H 76 units3
20.
Bonus Find the cross product of v and -2w if
v = 〈2, 4, −1〉 and w = 〈−1, 2, -5〉.
Chapter 8
38
J 88 units3
B:
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
15. An airplane takes off in the direction of the vector 〈9, 5〉. What
is the measure of the angle the plane makes with the horizontal?
A 29.1°
B 33.7°
C 56.3°
D 60.9°
NAME
DATE
8
PERIOD
Chapter 8 Test, Form 2A
SCORE
Write the letter for the correct answer in the blank at the right of each question.
1. Find the vertical component of v with a magnitude of 89.7 feet and a
direction angle of 12.8°.
A 19.38 ft
B 19.87 ft
C 87.58 ft
D 887.47 ft
1.
2. Find a unit vector u with the same direction as v = 〈−2, 4〉.
〈- −25 , −25 〉
√
√
G 〈4, −2〉
H
2 5
〈- −55 , −
5 〉
√
√
J
2 5
〈−55 , - −
5 〉
√
√
2.
3. Find the resultant of the sum of the pair of vectors using either the
triangle method or the parallelogram method. State the magnitude
in centimeters.
A 3.73 cm; 55.4°
C 5.0 cm; 55.4 °
B 3.73 cm; 90°
D 5.0 cm; 90°
3.
a
b
4. Find the measure of the angle θ between vectors a = 〈5, 6〉 and b = 〈−2, 8〉
to the nearest tenth of a degree.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
F 53.8°
G 36.2°
H 30.5°
J 28.5°
4.
⎯⎯⎯⎯ with initial point
5. Find the component form and magnitude of AB
A(9, 2) and terminal point B(−6, 3).
A 〈-15, 1〉; 15.03
B 〈3, 5〉; 5.83
C 〈15, -1〉; 3.74
D 〈3, 1〉; 3.16
5.
6. A force F1 of 12 newtons pulls due north. A force F2 of 5 newtons pulls
due east. Find the magnitude and direction of the resultant force.
F 13 N; 22.6°
G 17 N; 22.6°
H 13 N; 67.4°
J 17 N; 67.4°
6.
For Questions 7 and 8, find each of the following for
v = 〈−4, 0〉, w = 〈−3, 4〉, r = 〈−3, 7, 2〉, and s = 〈6, −3, 5〉.
7. 4w − 2v
A 〈−20, 16〉
B 〈−4, 16〉
C 〈−10, −8 〉
D 〈−22, 8〉
7.
G 〈15, 13, 12〉
H 〈−15, 13, −8〉
J 〈−9, 10, −3〉
8.
8. r – 2s
F 〈9, 1, −8〉
9. Arjon pushes a box up a ramp with a constant force of 45.8 newtons at a
constant angle of 55°. Find the work done in joules to move the box 8 meters.
A 183.8 J
B 210.2 J
C 300.1 J
D 523.3 J
9.
10. Find the cross product of v = 〈-9, 4, -8〉 and w = 〈6, −2, 4〉.
F 〈−54, −8, −32〉
Chapter 8
G 〈0, −12, −6〉
H 〈32, 84, 42〉
39
J 〈−6, −12, 0〉
10.
Glencoe Precalculus
Assessment
F
NAME
DATE
8
Chapter 8 Test, Form 2A
PERIOD
(continued)
11. Find the measure of the angle θ between u = 〈3, −2, 0〉 and v = 〈−4, 3, 1〉
to the nearest tenth of a degree.
A 11.7°
B 109.0°
C 168.3°
11.
D 176.8°
For Questions 12 and 13, find each dot product. Then determine
if the vectors are orthogonal.
12. a · b for a = −8i + 3j and b = 4i + 6j
F −50, not orthogonal
G 0, orthogonal
〈
〉 〈2
H −14, not orthogonal
J 21, not orthogonal
12.
C 5, orthogonal
D 0, not orthogonal
13.
〉
5
3
1
1
, −−
· −
, −2, − −
13. 4, −
4
3
2
A 5, not orthogonal
B 0, orthogonal
14. A constant force of 18 newtons is being applied at a constant angle of 56°
on an object at the same time that a constant force of 32 newtons at a
constant angle of 124° is acting on the object. What is the magnitude
and direction of the resultant force?
F 42.2 N; 100.7°
G 42.2 N; 280.7°
H 44.6 N; 36.5°
J 44.6 N; 216.5° 14.
15. An airplane is traveling due east with a velocity of 550 miles per hour.
The wind blows at 80 miles per hour at an angle of 40° North of East.
Determine the velocity of the airplane’s flight.
B 611.3 mph
C 613.4 mph
D 630 mph
15.
16. A cruise ship’s path is represented by the vector 〈−2, 12〉. It then follows
a new path represented by the vector 〈7, 6〉. What is the resultant path?
F 〈9, 6〉
G 〈9, 18〉
H 〈5, 6〉
J 〈5, 18〉
16.
17. Find the projection of u = 〈−4, 8〉 onto v = 〈3, 12〉.
A
28 112
,−
〈−
17 17 〉
B
17 17
,−
〈−
28 112 〉
C
112 224
,−
〈- −
51 51 〉
D
51 51
,−
〈- −
112 224 〉
17.
18. Find the area of the parallelogram with adjacent sides u = 〈23, 14, −28〉
and v = 〈12, 16, 13〉.
F 200 units2
G 630 units2
H 635 units2
J 916.6 units2
18.
19. The position of one airplane is represented by 〈−9, 8, 2.5) and a second
airplane is represented by 〈12, 2, 5). Determine the distance between
the planes if one unit represents one mile.
A 22.0 mi
B 38.5 mi
C 45.8 mi
D 56.7 mi
19.
J 230 units3
20.
20. Find the volume of the parallelepiped with adjacent edges
u = 〈1, −4, 2〉, v = 〈6, −5, 1〉, and w = 〈3, −4, −8〉.
F 90 units3
G 126 units3
H 178 units3
3
1
Bonus Find the cross product of - −
v and −
w if
2
4
v = 〈−2, 12, −3〉 and w = 〈−7, 4, -6〉.
Chapter 8
40
B:
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A 601.4 mph
NAME
DATE
8
PERIOD
Chapter 8 Test, Form 2B
SCORE
Write the letter for the correct answer in the blank at the right of each question.
1. Find the vertical component of v with a magnitude of 6.1 inches and
a direction angle of 55°.
A 3.50 in.
B 5.00 in.
C 7.45 in.
D 10.64 in.
1.
F
〈- −25 , −25 〉
√
√
G 〈−6, −3〉
H
〈- −, −〉
√5
2 √5
5
5
J
2 5
〈- −55 , - −
5 〉
√
√
2.
3. Find the resultant of the pair of vectors using either the triangle method or the
parallelogram method. State the magnitude of the resultant in centimeters.
A 2.5 cm; 50.8°
C 2.8 cm; 50.8°
B 2.5 cm; 39.2°
D 2.8 cm; 39.2°
a
3.
b
4. Find the measure of the angle θ between the vectors
a = 〈−15, 4〉 and b = 〈3, 10〉 to the nearest tenth of a degree.
F 62.4°
G 71.6°
H 87.1°
J 91.8°
4.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
⎯⎯⎯⎯ with initial point A(−4.3, −0.9)
5. Find the component form and magnitude of AB
⎯⎯⎯⎯.
and terminal point B(−2.8, 0.2). Then find the magnitude of AB
A 〈1.5, 1.1〉; 3.46
C 〈1.5, 1.1〉; 1.86
B 〈−7.1, −0.7〉; 7.13
D 〈−7.1, −1.1〉; 7.18
5.
6. A force F1 of 6 newtons pulls due north. A force F2 of 8 newtons pulls
due east. Find the magnitude and direction of the resultant force.
F 10 N; 53.1°
G 14 N; 53.1°
H 10 N; 36.9°
J 14 N; 36.9°
6.
For Questions 7 and 8, find each of the following for
v = 〈−4, 0〉, w = 〈−2, 4〉, r = 〈−3, 7, 2〉, and s = 〈6, −3, 5〉.
7. 3w − 4v
A 〈−22, 12〉
B 〈−6, 0〉
C 〈−10, −12 〉
D 〈10, 12〉
7.
G 〈−12, 17, −1〉
H 〈−15, 1, −8〉
J 〈0, 11, 9〉
8.
8. 2r – s
F 〈9, 1, −8〉
9. Jerard pushes a box up a ramp with a constant force of 55.2 newtons at a
constant angle of 22°. Find the work done in joules to move the box 4 meters.
A 82.7 J
B 89.2 J
C 157.3 J
D 204.7 J
9.
10. Find the cross product of v = 〈-5, 4, -8〉 and w = 〈6, -2, 6〉.
F 〈34, -78, 34〉
Chapter 8
G 〈-8, 16, 14〉
H 〈8, -16, -14〉
41
J 〈-34, 78, -34〉 10.
Glencoe Precalculus
Assessment
2. Find a unit vector u with the same direction as v = 〈−3, −6〉.
NAME
DATE
8
Chapter 8 Test, Form 2B
PERIOD
(continued)
11. Find the measure of the angle θ between u = 〈−3, −2, 1〉 and v = 〈−4, 3, 0〉
to the nearest tenth of a degree.
A 15.8°
B 71.3°
C 108.7°
D 164.2°
11.
For Questions 12 and 13, find each dot product. Then determine if
the vectors are orthogonal.
12. a · b for a = 8i + 3j and b = −4i − 6j
F −50, not orthogonal
G 1, orthogonal
H −14, orthogonal
J 13, orthogonal
12.
C −32, orthogonal
D −32, not orthogonal
13.
13. 〈4, −2, −2〉 · 〈-7, −2, 4〉
A 0, orthogonal
B −40, not orthogonal
14. A constant force of 22 newtons is being applied at a constant angle of 48°
on an object at the same time that a constant force of 54 newtons is being
applied at a constant angle of 135°. What is the magnitude and direction
of the resultant force?
F 56.5 N, 66.7°
G 59.3 N, 113.3°
H 56.5 N, 113.3°
J 59.4 N, 66.7° 14.
15. An airplane is traveling due east with a velocity of 620 miles per hour.
The wind blows at 45 miles per hour at an angle of 30° North of East.
Determine the velocity of the airplane’s flight.
B 636.6 mph
C 659.4 mph
D 665 mph
15.
16. A cruise ship’s path is represented by the vector 〈8, −2〉. It then follows a
new path represented by the vector 〈−2, −8〉. What is the resultant path?
F 〈10, −12〉
G 〈10, −4〉
H 〈6, −10〉
J 〈6, −4〉
16.
17. Find the projection of u = 〈3, −12〉 onto v = 〈9, 2〉.
A
27 6
,−
〈−
85 85 〉
B
85 85
,−
〈−
27 6 〉
C
9
36
, -−
〈−
85
85 〉
D
85
85
, -−
〈−
9
36 〉
17.
18. Find the area of the parallelogram with adjacent sides
u = 〈6, −8, −31〉 and v = 〈53, 7, 21〉.
F 1727.0 units2
G 1785.8 units2
H 1830.0 units2
J 2132.4 units2 18.
19. The position of one airplane is represented by 〈12, 2, 3.7〉 and a second
airplane is represented by 〈−1, 7, 5.2〉. Determine the distance between
the planes if one unit represents one mile.
A 14.0 mi
B 16.8 mi
C 26.6 mi
D 48.9 mi
19.
J 236 units3
20.
20. Find the volume of the parallelepiped with adjacent edges
u = 〈1, 2, 2〉, v = 〈6, −5, 1〉, and w = 〈6, −4, −8〉.
F 68 units3
G 132 units3
H 164 units3
Bonus Find the cross product of 3v and -2w if
v = 〈−1, 5, 3〉 and w = 〈−7, 5, -6〉.
Chapter 8
42
B:
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A 536.9 mph
NAME
DATE
8
Chapter 8 Test, Form 2C
PERIOD
SCORE
1. The vector v has a magnitude of 10 meters and a direction of
92°. Find the magnitude of its vertical and horizontal
components.
1.
2. Find a unit vector u with the same direction as v = 〈-3, 5〉.
2.
B
Assessment
3. Find a - 3b for the pair of vectors below using either the
triangle method or the parallelogram method. State the
magnitude of the resultant in centimeters and its direction
relative to the horizontal.
C
3.
4. Find the measure of the angle θ between u = 3i - 4 j
and v = i + 2 j.
4.
⎯⎯⎯⎯ with initial
5. Find the component form and magnitude of AB
point A(0, -8) and terminal point B(−1, 7).
5.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6. A force F1 of 27 newtons pulls at an angle of 23° above due east.
A force F2 of 33 newtons pulls at an angle of 55° below due west.
Find the magnitude and direction of the resultant force.
6.
For Questions 7 and 8, find each of the following for
v = 〈1, −6〉, w = 〈2, −5〉, r = 〈1, −1, 0〉, and s = 〈10, −6, 5〉.
7. v + 3w
7.
8. 3s - 2r
8.
9. Jerard pushes a box up a ramp with a constant force of
41.5 newtons at a constant angle of 28°. Find the work done
in joules to move the box 5 meters.
9.
10. Find the cross product of v = 〈6, −4, 3〉 and w = 〈4, 2, -6〉.
10.
11. Find the measure of the angle θ between u = 〈4, −2, 1〉
and v = 〈−3, 5, 0〉 to the nearest tenth of a degree.
11.
For Questions 12 and 13, find each dot product. Then
determine if the vectors are orthogonal.
12. 〈8, 2〉 · 〈0, −6〉
12.
13. 〈3, −7, 4〉 · 〈−4, −2, 1〉
13.
Chapter 8
43
Glencoe Precalculus
NAME
8
DATE
Chapter 8 Test, Form 2C
PERIOD
(continued)
14. A constant force of 15 newtons is being applied at a constant
angle of 30° on an object at the same time that a constant force
of 25 newtons at a constant angle of 60° is acting on the object.
What is the magnitude and direction of the resultant force?
14.
15. An airplane is traveling due east with a velocity of 620 miles per
hour. The wind blows at 60 miles per hour at an angle of 50° with
the horizontal. Determine the velocity of the airplane’s flight.
15.
16.
17. Find the projection of u = 〈7, -2〉 onto v = 〈−1, 5〉.
17.
18. Find the surface area of the parallelepiped with adjacent edges
u = 〈8, 3, −3〉 and v = 〈−8, 9, 13〉, and w = 〈7, −2, −7〉.
18.
19. The position of one airplane is represented by 〈9, 13, 4〉 and a
second airplane is represented by 〈−7, 12, 3〉. Determine the
distance between the planes if one unit represents one mile.
19.
20. Find the volume of the parallelepiped with adjacent edges
u = 〈0.5, −3, 2〉, v = 〈−7, −0.5, 1〉, and w = 〈−4, 7, −8〉.
20.
Bonus If v = 〈v1, v2〉, where v1 and v2 are not both 0 and u = −3v,
find the measure of the angle between u and v.
Chapter 8
44
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
16. A cruise ship’s path travels for 24.7 miles at an angle of 58.2°.
It then follows a path can be represented by the vector
〈22, 31〉. What is the resultant path?
B:
Glencoe Precalculus
NAME
DATE
8
Chapter 8 Test, Form 2D
PERIOD
SCORE
1. The vector v has a magnitude of 5 meters and a direction of
60°. Find the magnitude of its vertical and horizontal
components.
1.
2. Find a unit vector u with the same direction as v = 〈−3, 8〉.
2.
B
Assessment
1
3. Find −−
a + b for the pair of vectors below using either the
2
triangle method or the parallelogram method. State the
magnitude of the resultant in centimeters and its direction
relative to the horizontal.
C
3.
4. Find the measure of the angle θ between u = 6i - 3 j and
v = 2i + j to the nearest tenth of a degree.
4.
⎯⎯⎯⎯ with initial
5. Find the component form and magnitude of AB
point A(3, −1) and terminal point B(−1, −2).
5.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6. A force F1 of 25 newtons pulls at an angle of 20° above due east.
A force F2 of 35 newtons pulls at an angle of 55° below due west.
6.
Find the magnitude and direction of the resultant force.
For Questions 7 and 8, find each of the following for
v = 〈2, −3〉, w = 〈1, 5〉, r = 〈1, −1, 1〉, and s = 〈0, −3, 2〉.
7. −2v + w
7.
8. 3s + r
8.
9. Harold pushes a box up a ramp with a constant force of
31.5 newtons at a constant angle of 32°. Find the work
done in joules to move the box 8 meters.
9.
10. Find the cross product of v = 〈2, −1, 3〉 and w = 〈1, 0, −5〉.
10.
11. Find the measure of the angle θ between u = 〈−2, −2, 1〉
and v = 〈−3, 4, 0〉 to the nearest tenth of a degree.
11.
For Questions 12 and 13, find each dot product. Then
determine if the vectors are orthogonal.
12. 〈2, 0〉 · 〈0, −5〉
12.
13. 〈3, −4, −2〉 · 〈−2, −2, 3〉
13.
Chapter 8
45
Glencoe Precalculus
NAME
8
DATE
Chapter 8 Test, Form 2D
PERIOD
(continued)
14.
15. An airplane is traveling due east with a velocity of 580 miles per
hour. The wind blows at 50 miles per hour at an angle of 45° to
the horizontal. Determine the velocity of the airplane’s flight.
15.
16. A cruise ship’s path travels for 16.1 miles at an angle of 60.3°.
It then follows a path that can be represented by the vector
〈22, 7〉. What is the resultant path?
16.
17. Find the projection of u = 〈9, −5〉 onto v = 〈3,−2〉.
17.
18. Find the surface area of the parallelepiped with adjacent edges
u = 〈4, 7, −8〉, v = 〈−2, 5, 11〉, and w = 〈9, −2, −8〉.
18.
19. The position of one airplane is represented by 〈11, 10, 3〉 and a
second airplane is represented by 〈−9, 14, 3〉. Determine the
distance between the planes if one unit represents one mile.
19.
20. Find the volume of the parallelepiped with adjacent edges
u = 〈-2, 0.75, 4〉, v = 〈6, −0.3, 8〉, and w = 〈3, −2.5, 9〉.
20.
Bonus If v = 〈v1, v2〉 where v1 and v2 are not both 0 and u = 2v,
find the measure of the angle between u and v.
B:
Chapter 8
46
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
14. A constant force of 10 newtons is being applied at a constant
angle of 45° on an object at the same time that a constant force
of 20 newtons at a constant angle of 65° is acting on the object.
What is the magnitude and direction of the resultant force?
Glencoe Precalculus
NAME
DATE
8
PERIOD
Chapter 8 Test, Form 3
SCORE
1. The vector v has a magnitude of 11.4 meters and a direction of
248°. Find the magnitude of its vertical and horizontal
components.
1.
2. Find a unit vector u with the same direction as v = 〈−6, −2〉.
2.
B
Assessment
1
2
3. Find −3a + −
b+−
a for the pair of vectors below using
2
3
either the triangle method or the parallelogram method. State
the magnitude of the resultant in centimeters and its direction
relative to the horizontal.
C
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3.
4. Find the measure of the angle θ between u = 4i - 8j
and v = 2i - 5j.
4.
⎯⎯⎯⎯ with initial
5. Find the component form and magnitude of AB
point A(1.8, −3.8) and terminal point B(−0.1, −3.8).
5.
6. A force F1 of 18.8 newtons pulls at an angle of 12° north of
east. A force F2 of 3.2 newtons pulls at an angle of 55° south of
east. Find the magnitude and direction of the resultant force.
6.
For Questions 7 and 8, find each of the following for
〈
〈 2〉
〉
〈
〈
〉
〉
3
3
1
1
, w = 2, - −
, r = 1, − −
, 2 , and s = 10, −6, −
.
v = 0, −
4
4
4
1
7. −v + −
w
7.
1
8. −
r + 4s
8.
9. Kyle is pulling a box east with a force of 300 newtons at a
constant angle of 42° to the horizontal. Jerome is pushing the
box from behind with a force of 350 newtons due east.
Determine the magnitude and direction of the resultant force
on the box.
9.
3
2
〈
〉
〈
〉
1
1
10. Find the cross product of v = 6, − −
, 3 and w = 4, 2, − −
.
2
〈
3
1
,4
11. Find the measure of the angle θ between u = −4, − −
〈
〉
1
.
and v = −3, −4, −
Chapter 8
3
47
2
〉
10.
11.
Glencoe Precalculus
NAME
8
DATE
Chapter 8 Test, Form 3
PERIOD
(continued)
For Questions 12 and 13, find each dot product. Then
determine if the vectors are orthogonal.
〈 3〉 〈2
2
1
12. 8, −
· −
, −6
〉
12.
〈
〉
1
13. 〈−2, 6, 8〉 · −4, −2, − −
2
13.
14. A constant force of 12.2 newtons is being applied at a constant
angle of 12° on an object at the same time that a constant force of
18.9 newtons at a constant angle of 75.8° is acting on the object.
What is the magnitude and direction of the resultant force?
14.
15. An airplane is traveling with a velocity of 300 miles per hour
at an angle 45° to the north of east. If a 40-mile per hour wind
is blowing from a bearing of 130°, determine the velocity and
direction of the airplane relative to the ground.
15.
〈
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
16. A cruise ship starts on a path represented by the vector
〈9, -10〉. It then changes direction and the path can be
represented by the vector 〈3, -13〉. Another ship’s path is
represented by the path 〈14, -4〉. If the second ship changes
direction, what vector represents the path the second ship
would need to take to end up at the same place as the first ship? 16.
〉
3
17. Find the projection of u = −
, −4 onto a vector v with a length
2
17.
of 2.8 units and a direction angle of 35°.
18. Find the surface area of the parallelepiped with adjacent edges
1
1
u = 〈3, 5, 9〉 and v = 3, −
, 8 and w = 2, −4, −
.
18.
19. Two airplanes are flying to the same airport. The position of the
first airplane is (5, -9, 2.5) and the position of the second
airplane is (-8, 7, 2). The airport is at (3, 3, 0.5). Which airplane
is closer to the airport if the positions are given in miles?
How much closer?
19.
20. Find the volume of the parallelepiped with adjacent edges
u = −3i + 2.75j + k, v = 4i −1.3j + 8k,
and w = −3i − 0.5j + 2k.
20.
Bonus If v = 〈v1, v2〉 where v1 and v2 are not both 0 and u = -2v
and w = 5v, find the measure of the angle between u and w.
B:
〈
Chapter 8
4
〉
〈
2
48
〉
Glencoe Precalculus
NAME
DATE
8
Extended-Response Test
PERIOD
SCORE
Demonstrate your knowledge by giving a clear, concise solution to
each problem. Be sure to include all relevant drawings and justify
your answers. You may show your solution in more than one way
or investigate beyond the requirements of the problem.
1. Given the vectors below, complete the questions that follow.
30°
C
30°
c = 〈-3, 1〉 and d = 〈-8, -11〉
Assessment
B
a. Show two ways to find a + b.
b. Find a - b. Explain each step.
c. Does a + b = b + a? Why or why not?
d. Does a - b = b - a? Defend your answer.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
e. Tell how to find the sum c + d. Find the sum and its magnitude.
f. Find two vectors whose difference is 〈4, -1, 3〉. Write the difference as
a linear combination.
g. Find a vector perpendicular to 〈7, -3〉. Explain how you know that
the two vectors are perpendicular.
h. Find a × b if a = 〈2, 1, 0〉 and b = 〈1, 3, 0〉. Graph the vectors and
the cross product c in three dimensions.
2. An airplane can be represented by the point (9, 8, 5). The airport can be
represented by the point (8, -3, 0).
a. How far away from the airport is the plane?
b. A second plane can be represented by the point (10, -6, 4).
How far away from the airport is the second plane?
c. If the first plane is moving at 200 miles per hour and the second
plane is moving at 150 miles per hour, which plane will be at the
airport first?
3. Find two pairs of perpendicular vectors. Then verify that they are
perpendicular by calculating their dot products.
Chapter 8
49
Glencoe Precalculus
NAME
DATE
PERIOD
8
Standardized Test Practice
SCORE
(Chapters 1–8)
Part 1: Multiple Choice
Instructions: Fill in the appropriate circle for the best answer.
1. Find the period and phase shift of the graph of f (x) = -4 cos (2x + π).
A 2, 4 right
B 2π, π left
π
left
C π, −
D 2, π right
H 1
J −
2
1.
A
B
C
D
√3
2
2.
F
G
H
J
D 17
3.
A
B
C
D
1
J f-1(x) = - −
4.
F
G
H
J
3π
2. Find the value of sin −
.
2
√3
2
G −1
F -−
3. Find the cross product of u = 〈8, -3, 9〉 and v = 〈−2, 3, 2〉.
B 〈-33, -34, 18〉 C −7
A 〈-33, 34, 18〉
4. Find f -1(x) if f (x) = −8x - 3.
x+3
8
F f-1(x) = −
x+3
8
1
G f-1(x) = - − H f-1(x) = −
8x + 3
8x + 3
5. The graph of g(x) = (x + 3)3 + 4 can be obtained from the graph of
f (x) = x3 by performing which transformation?
C move right 3 and down 4
B move left 3 and up 4
D move left 3 and down 4
5.
A
B
C
D
H 10
J 64
6.
F
G
H
J
D 〈-11, 12〉
7.
A
B
C
D
8.
F
G
H
J
9.
A
B
C
D
10.
F
G
H
J
4 -2 1
6. Find the value of 0 8 -4 .
2 2 -1
F -16
G 0
7. Let u = 〈5, 3〉 and v = 〈-7, 2〉. Find 2u + 3v.
A 〈10, 6〉
B 〈−21, 6〉
C 〈−11, 6〉
8. If f(x) = -3x + 7 and g(x) = 2x - 4, find f (g(x)).
F f (g(x)) = -6x + 12
H f (g(x)) =−6x + 10
G f (g(x)) = −6x + 19
J f (g(x)) = −6x + 14
⎡ 9 -2⎤ ⎡ 3 1 ⎤
9. Find the element in row 1, column 2 of the product ⎢
.
⎢
⎣-1 5⎦ ⎣ 2 -4 ⎦
A −2
B -1
C 7
D 17
G 6
H 8
J 4096
10. Find log2 64.
F 2
Chapter 8
50
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A move right 3 and up 4
NAME
DATE
8
Standardized Test Practice
PERIOD
(continued)
(Chapters 1–8)
11. Solve log3 x = 5.
A 3
B 8
C 15
D 243
11.
A
B
C
D
12.
F
G
H
J
D the line y = x 13.
A
B
C
D
14.
F
G
H
J
15.
A
B
C
D
16.
F
G
H
J
A relative maximum of −7 at x = 1
C relative maximum of 0 at x = -6
B relative minimum of −7 at x = 1
D relative minimum of 0 at x = −6 17.
A
B
C
D
18.
F
G
H
J
19.
A
B
C
D
F
G
H
J
F 8
G 32
H 22
J 40
13. The graph of f (x) = 2x3 + 17x is symmetric with respect to
which of the following?
A the x-axis
B the y-axis
C the origin
14. Given that f (x) = 4x2 - 3x + 2, what is the value of f (−3)?
F −25
G 12
H 29
J 47
15. What are the zeros of f (x) = 2x2 + 3x - 5?
A -1, 5
B −5, 1
5
C -−
,1
2
5
D −1, −
2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
16. Choose the statement that is true for the graph of f (x) = −2(x - 3)4.
F f (x) increases for x > -3.
H f (x) increases for x > 3.
G f (x) decreases for x < -3.
J f (x) decreases for x > 3.
17. Which is true for the graph of f (x) = 2x3 - 3x2 - 6?
⎧
3x - 9 if x < -3
18. Evaluate g(3) for g(x)= 2x2 - 15 if 0 < x < 2.
⎩ −x + 12 if x > 2
⎨
F 0
G 3
H 9
J 15
19. The height of a softball thrown straight up into the air can be modeled
by the function h(t) = −4.9t2 + 4t + 2, where t is the time in seconds
after the ball is released and h(t) is the height of the ball in meters.
What is the highest point the ball reaches?
A 2.0 m
B 2.8 m
C 3.1 m
D 4.5 m
20. Marina deposited $8000 in an account paying 4.5% interest compounded
continuously. How long would it take for the balance in the account to double?
F 9.0 yr
Chapter 8
G 15.4 yr
H 20.0 yr
51
J 27.6 yr
20.
Glencoe Precalculus
Assessment
12. What is the remainder of f (x) = 3x3 + 4x2 - 8 divided by x - 2?
NAME
DATE
8
Standardized Test Practice
PERIOD
(continued)
(Chapters 1–8)
Part 2: Short Response
Instructions: Write your answers in the space provided.
π
,a
21. Write an equation of the sine curve that has a period of −
3
vertical shift of 9, and an amplitude of 3.
21.
22. Find the equations of the vertical asymptote(s) of the
3x2 + 5x - 6
x-5
22.
function f(x) = − .
23. An athlete is running at 4.6 meters per second and throws a
javelin at 8 meters per second at an angle of 50°. What is the
23.
resulting speed and direction of the javelin?
⎡6 8 x⎤ ⎡-1 4⎤ ⎡ 18 70⎤
24.
24. Find the value of x and y. 1 -2 1
3 y = -7 -9
⎣0 3 6⎦ ⎣ 0 -3⎦ ⎣ 9 -3⎦
⎢
⎢
⎢
25. A horse is pulling a wagon with a constant force of 1500 N.
The harness makes an angle of 28° with the wagon.
Find the work done in joules to pull the wagon 50 meters.
25.
26. What is the angle between the vectors u = 〈2, −3, 4〉
and v = 〈9, 0, -1〉?
26.
"
$
#
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
27. If AB = 22.5 and AC = 13.8, find
CB and the measure of each angle.
27.
28. Mary is in a canoe crossing a river that has a current flowing
at 3.2 meters per second due east. Mary begins paddling across
the river at 3.4 meters per second at an angle of 70° with the
bank. Find the magnitude and direction of the canoe’s path. 28.
29. Write an equation for a circle that has its center at (9, −2)
and a radius of 5 units.
29.
30. Two airplanes are flying to an airport that can be represented
by the point (30, 22, 1). The first airplane’s position can be
represented by the point (-15, 4, 2.5) and the second plane’s
position can be represented by the point (43, -6, 3). Each
unit represents 1 mile.
a. What vector represents the direct path from the first
plane to the airport?
30a.
b. What vector represents the direct path from the second
plane to the airport?
30b.
c. Which plane is closer to the airport, the first plane or
the second plane? How much closer?
30c.
Chapter 8
52
Glencoe Precalculus
Chapter 8
A1
Glencoe Precalculus
Before you begin Chapter 8
Vectors
Anticipation Guide
DATE
PERIOD
After you complete Chapter 8
12. The area of a parallelogram can be found using dot products.
10. Vectors can be used to represent forces in three-dimensional
space.
11. When finding torque, the dot product is used.
6. Any vector can be broken down into horizontal and vertical
components.
7. To add two vectors algebraically, add all of the numbers in
the first vector together. Then add all of the numbers in the
second vector together.
8. Trigonometric ratios sometimes need to be used when
working with vectors.
9. The dot product of two vectors is a scalar.
5. When vectors are combined the result is a scalar.
3. The initial point of a vector is the point where the vector
starts.
4. Vectors can be used to represent forces applied at an angle.
2. Vectors can be represented by directed line segments.
1. Scalars have both magnitude and direction.
Statement
3
10/23/09 11:05:00 AM
Glencoe Precalculus
PERIOD
0 Y
125°
8
0
4
X
&
1 cm: 20 mi/h
45°
/
Using a scale of 1 cm.: 20 mi/h, draw
and label a 55 ÷ 20 or 2.75-centimeter
arrow 45° east of south.
b. w = 55 miles per hour at a bearing
of S45°E
45°
0
1 cm: 50 yd
40°
U
005_032_PCCRMC08_893809.indd 5
Answers
Chapter 8
3.
2.2 cm, 140°
C
B
S
5
4.
F
G
F
Lesson 8-1
10/23/09 11:17:57 AM
Glencoe Precalculus
3.5 cm, -12°
S
G
2. t = 150 yards at 40° to the horizontal
Find the resultant of each pair of vectors using either the triangle or
parallelogram method. State the magnitude of the resultant in centimeters
and its direction relative to the horizontal.
8 1 cm: 10 m
S
/
1. r = 30 meters at a bearing of N45°W
Use a ruler and a protractor to draw an arrow diagram for each quantity
described. Include a scale on each diagram.
Exercises
1 cm: 20 lb
W
Z
Using a scale of 1 cm: 20 lb, draw and
label a 60 ÷ 20 or 3-centimeter arrow
in standard position at a 125° angle to
the x-axis.
a. v = 60 pounds of force at 125° to
the horizontal
Example
Use a ruler and a protractor to draw an arrow diagram for each
quantity described. Include a scale on each diagram.
A vector is a quantity that has both magnitude and direction. The
magnitude of a vector is the length of a directed line segment, and the direction of a vector
is the directed angle between the positive x-axis and the vector. When adding or subtracting
vectors, you can use the parallelogram or triangle method to find the resultant.
Introduction to Vectors
C
Chapter 8
DATE
Study Guide and Intervention
Geometric Vectors
8-1
NAME
• For those statements that you mark with a D, use a piece of paper to write an example
of why you disagree.
Chapter Resources
B
D
D
A
A
A
D
A
A
D
A
D
A
STEP 2
A or D
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
• Did any of your opinions about the statements change from the first column?
• Reread each statement and complete the last column by entering an A or a D.
Step 2
STEP 1
A, D, or NS
• Write A or D in the first column OR if you are not sure whether you agree or disagree,
write NS (Not Sure).
• Decide whether you Agree (A) or Disagree (D) with the statement.
• Read each statement.
Step 1
8
NAME
0ii_004_PCCRMC08_893809.indd Sec1:3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers (Anticipation Guide and Lesson 8-1)
(continued)
PERIOD
60°
x
50 N
y
A2
Use a calculator.
⎪x⎥ = 25
⎪y ⎥
⎪y⎥ ≈ 43.3
⎪y⎥ = 50 sin 60°
50
sin 60° = −
120°
8
50°
2.5 cm
4
/
&
2. 2.5 centimeters per hour at a bearing of
N50°W -1.9, 1.6
Glencoe Precalculus
005_032_PCCRMC08_893809.indd 6
Chapter 8
6
Glencoe Precalculus
4. TRANSPORTATION A helicopter is moving 15° north of east with a velocity of 52 km/h.
If a 30-kilometer per hour wind is blowing from a bearing of 250°, find the helicopter’s
resulting velocity and direction. 81.93 km/h; 16.8° north of east
3. YARDWORK Nadia is pulling a tarp along level ground with a force of 25 pounds
directed along the tarp. If the tarp makes an angle of 50° with the ground, find the
horizontal and vertical components of the force. What is the magnitude and direction
of the resultant? 16.07 lb; 19.15 lb; 25 lb; 50°
7 in.
1. 7 inches at a bearing of 120°
from the horizontal -3.5, 6.1
Draw a diagram that shows the resolution of each vector into its rectangular
components. Then find the magnitudes of the vector’s horizontal and vertical
components.
Exercises
The magnitude of the horizontal component is about 25 Newtons, and the
magnitude of the vertical component is about 43 Newtons.
Solve for x and y.
Right triangle definitions of cosine and sine
⎪x⎥ = 50 cos 60°
50
cos 60° = −
⎪ x⎥
The horizontal and vertical components of the force form a right triangle.
Use the sine or cosine ratios to find the magnitude of each force.
b. Find the magnitudes of the horizontal and vertical components of the force.
Jamal’s pull can be resolved into a horizontal
pull x forward and a vertical pull y upward
as shown.
a. Draw a diagram that shows the resolution of the force Jamal exerts into its
rectangular components.
Example
Suppose Jamal pulls on the ends of a rope tied to a dinghy with a
force of 50 Newtons at an angle of 60° with the horizontal.
Vectors can be resolved into horizontal and vertical components.
Introduction to Vectors
Study Guide and Intervention
DATE
10/23/09 11:25:56 AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 8
Vector Applications
8-1
NAME
Introduction to Vectors
Practice
DATE
PERIOD
4
45°
1 cm: 20 m
S
&
Z
U
1 cm: 25 lb
60°
Y
2. t = 100 pounds of force at 60° to the
horizontal
8
1000
45°
4
/
&
Y
Z
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC08_893809.indd 7
Chapter 8
bearing of 354.3°
7
Lesson 8-1
10/23/09 11:09:29 AM
Glencoe Precalculus
5. AVIATION An airplane is flying with an airspeed of 500 miles per hour on a heading
due north. If a 50-mile per hour wind is blowing at a bearing of 270°, determine
the velocity and direction of the plane relative to the ground. 502.49 mph;
44.6 N; 40.1 N
b. Find the magnitudes of the horizontal and vertical components of the force.
42°
60 N
a. Draw a diagram that shows the resolution of the force Roland exerts into
its rectangular components.
4. CONSTRUCTION Roland is pulling a crate of construction materials with a
force of 60 Newtons at an angle of 42° with the horizontal.
200
1150 ft at a bearing of 52.1° north of west
3. GROCERY SHOPPING Caroline walks 45° north of west for 1000 feet and
then walks 200 feet due north to go grocery shopping. How far and at what
north of west quadrant bearing is Caroline from her apartment?
8
/
1. r = 60 meters at a bearing of N45°E
Use a ruler and a protractor to draw an arrow diagram for each quantity
described. Include a scale on each diagram.
8-1
NAME
Answers (Lesson 8-1)
Chapter 8
4
150°
&
A3
PERIOD
4
35 m
20°
&
30°
45 m
79.7 m; E25.6°N
Glencoe Precalculus
b. Find the resulting distance and the
direction of her path.
8
/
a. Use a ruler and protractor to draw
an arrow diagram representing the
situation.
6. SKATEBOARDING Meredith is
skateboarding along a path 20° north
of east for 35 meters. She then changes
paths and travels for 45 meters along a
path 30° north of east.
119.5 meters; N55.1°W
5. ORIENTEERING In an orienteering
competition, Jada walks N70°W for
200 meters. She then walks due east for
90 meters. How far and at what bearing
is Jada from her starting point?
decrease
4. TRAVEL Karrie is pulling her luggage
across the airport floor. She applies a
22-newton force to the handle of the
bag when the bag makes a 72-degree
angle with the floor. What is the
magnitude of the force that moves the
luggage straight forward? What effect
would it have if Karrie moved the
handle closer to the floor, decreasing
the angle? 20.9 N; the force would
3. CANOEING A person in a canoe wants
to cross a 65-foot-wide river. He begins
to paddle straight across the river at
1.2 m/s while a current is flowing
perpendicular to the canoe. If the
resulting velocity of the canoe is 3.2 m/s,
what is the speed of the current to the
nearest tenth? 3.0 m/s
DATE
10/23/09 11:27:09 AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Precalculus
005_032_PCCRMC08_893809.indd 8
Chapter 8
No. Sample answer: If you use
the triangle method of vector
addition, the resultant will be in
the same position regardless of
the individual angles.
c. Would changing the angle of the
tractors affect the magnitude of the
resulting force if the angle between
the tractors remained 40°? Explain.
b. What is the resulting force on the
tree stump? 3293 N
3288.9 N; 171 N
a. What is the sum of the horizontal
components of the tractors? What is
the sum of the vertical components?
1500 N
20°
20°
2000 N
2. FARMING Two tractors are removing a
tree stump as shown. One tractor pulls
with a force of 2000 newtons, and the
other tractor pulls with a force of
1500 newtons. The angle between the
two tractors is 40°.
1 cm: 50 km
8
/
8
Introduction to Vectors
Word Problem Practice
1. SAILING A captain sails a boat east for
200 kilometers at a bearing of 150°. Use
a ruler and protractor to draw an arrow
diagram for the quantities described.
Include a scale on the diagram.
8-1
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Enrichment
DATE
PERIOD
180°
150°
70°
80 N
80 N
120 N
Answers
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC08_893809.indd 9
Chapter 8
9
2. MOVING Three men are trying to move a sofa. One man is pushing
on the sofa with a force of 40 N at an angle of 50° with the ground.
A second man exerts a force of 100 N at 110°, and a third man exerts
a force of 10 N at 150°. Find the magnitude and direction of the
resultant force. 130.7 N; 97.7°
1. DOGS Three dogs are pulling a wagon. One dog acts with 40 N at 50°
on the wagon. The second dog acts with 100 N at 110°. The third dog
acts with 10 N at 150°. Find the magnitude and direction of the
resultant force. 131 N; 98°
Find the magnitude and direction of the resultant force
acting on each object.
r
Lesson 8-1
3/25/09 10:04:51 PM
Glencoe Precalculus
a
Now add the resulting vector
to the 120 N vector.
The resultant force is 219 N at a direction of 145°.
100 N
a
Add the 80 N vector and the
100 N vector first.
First, add two of the vectors. The order in which the vectors are added does
not matter.
120 N
100 N
Example
CONSTRUCTION Kendra is pulling on a box with a force
of 80 newtons at an angle of 70° with the ground at the same time that
Kyle is pulling on the box with a force of 100 newtons at an angle of 150°
with the ground. A third force of 120 N acts at an angle of 180°. Find
the magnitude and direction of the resultant force acting on the box.
Three or more forces may work on an object at one time. Each of these forces
can be represented by a vector. To find the resultant vector that acts upon the
object, you can add the individual vectors two at a time.
More Than Two Forces Acting on an Object
8-1
NAME
Answers (Lesson 8-1)
Vectors in the Coordinate Plane
Study Guide and Intervention
DATE
(-4 - 2) 2 + [2 - (-3)] 2
√
A4
Subtract.
(x1, y1) = (2, −3) and (x2, y2) = (−4, 2)
Component form
= 〈4 + (-1), 2 + 3〉 or 〈3, 5〉
Vector addition
= 〈11, 9〉
−4
−2
2
0 2
y
X
4
x
2. A(-15, 0), B(7, -19) 〈22, -19〉, 13 √
5
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10
Glencoe Precalculus
6. f - 2(g + 2h) 〈-40, -32〉
5. 2g + h 〈47, 39〉
Chapter 8
4. 8g - 2f + 3h 〈181, 163〉
3. f - g 〈-20, -23〉
Find each of the following for f = 〈4, -2〉, g = 〈24, 21〉, and h = 〈-1, -3〉.
1. A(12, 41), B(52, 33) 〈40, -8〉, 8 √
26
⎯⎯⎯ with the
Find the component form and magnitude of the vector AB
given initial and terminal points.
Exercises
Scalar multiplication
Substitute.
Vector addition
Substitute.
= 〈12, 6〉 + 〈-1, 3〉
3s + t = 3〈4, 2〉 + 〈-1, 3〉
b. 3s + t
−4 −2
Y
4
Find each of the following for s = 〈4, 2〉 and t = 〈-1, 3〉.
s + t = 〈4, 2〉 + 〈-1, 3〉
a. s + t
Example 2
⎯⎯⎯ = 〈x2 - x1, y2 - y1〉
XY
= 〈-4 - 2, 2 - (-3)〉
= 〈-6, 5〉
⎯⎯⎯as an ordered pair.
Represent XY
= √
61 or about 7.8 units
(-6) 2 + 5 2
= √
=
⎪XY
⎯⎯⎯⎥ = √
(x2 - x1)2 + (y2 - y1)2
Example 1
⎯⎯⎯ with initial point X(2, −3) and
Find the magnitude of XY
terminal point Y(-4, 2).
⎯⎯⎯ using the Distance Formula.
Determine the magnitude of XY
The magnitude of a vector in the
coordinate plane is found using the Distance Formula.
PERIOD
10/24/09 10:52:30 AM
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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 8
Vectors in the Corrdinate Plane
8-2
NAME
DATE
(continued)
PERIOD
〉 〈
〉
⎯⎯⎯ be the vector with initial point M(2, 2) and terminal
Let MP
Scalar multiplication
Simplify.
〈a, b〉 = ai + bj
Component form
-3
〈−45 , −
5 〉
2. w = 〈10, 25〉
29
2 √
29 5 √
29
29
〈−, −〉
4. M(0, 6), N(18, 4) 18i - 2j
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005_032_PCCRMC08_893809.indd 11
Chapter 8
7. -4i + 2j 153.4°
8. 〈2, 17〉 83.3°
11
Lesson 8-2
10/23/09 11:43:12 AM
Glencoe Precalculus
6. |v| = 5, θ = 95° 〈-0.44, 5.0〉
Find the direction angle of each vector to the nearest tenth.
5. |v| = 18, θ = 240° 〈-9, -15.6〉
Find the component form of v with the given magnitude and direction angle.
3. M(2, 8), N(-5, -3) -7i - 11j
⎯⎯⎯⎯ be the vector with the given initial and terminal points. Write MN
⎯⎯⎯⎯
Let MN
as a linear combination of the vectors i and j.
1. p = 〈4, -3〉
Find a unit vector u with the same direction as the given vector.
Exercises
⎯⎯⎯⎯ = 〈3, 2〉
MP
= 3i + 2j
⎯⎯⎯⎯ = 〈x2 - x1, y2 - y1〉
MP
Component form
= 〈5 - 2, 4 - 2〉 or 〈3, 2〉
(x1, y1) = (2, 2) and (x2, y2) = (5, 4)
Then rewrite the vector as a linear combination of the standard unit vectors.
⎯⎯⎯ as a linear combination of the vectors i and j.
point P(5, 4). Write MP
⎯⎯⎯⎯.
First, find the component form of MP
Example 2
-4 -1
= −
, − or
√
17
17 √
〈
√
17
-4 √
17 - √
17
−, −
17
17
a2 + b2
⎪〈a, b〉⎥ = √
1
= −
〈-4, -1〉
√
(-4)2 + (-1)2
1
=−
〈-4, -1〉
Substitute.
Unit vector with the same direction as v
Find a unit vector u with the same direction as v = 〈-4, -1〉.
1
= −
〈-4, -1〉
⎪〈-4, -1〉⎥
1
u=−
v
⎪v⎥
Example 1
A vector that has a magnitude of 1 unit is called a unit vector.
A unit vector in the direction of the positive x-axis is denoted as i = 〈1, 0〉, and
a unit vector in the direction of the positive y-axis is denoted as j = 〈0, 1〉.
Vectors can be written as linear combinations of unit vectors by first writing
the vector as an ordered pair and then writing it as a sum of the vectors i and j.
Vectors in the Coordinate Plane
Study Guide and Intervention
Unit Vectors
8-2
NAME
Answers (Lesson 8-2)
Chapter 8
Vectors in the Coordinate Plane
Practice
DATE
PERIOD
10
〈1, -3〉; √
2. A(4, -2), B(5, -5)
A5
√
5 2 √5
5
5
〈- −, −〉
9. v = 〈-8, -2〉
4 √
17
17
√
17
17
〈- −, - −〉
146
〈11, 5〉; √
3. A(-3, -6), B(8, -1)
13. D(2, 1), E(3, 7) i + 6j
12. D(4, 6), E(-5, -2) -9i − 8j
〈8.9, 8.0〉
15. |v| = 8, θ = 132°
〈−5.4, 5.9〉
12
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Chapter 8
295.62 N
16. GARDENING Anne and Henry are lifting a stone statue and moving it to
a new location in their garden. Anne is pushing the statue with a force of
120 newtons at a 60° angle with the horizontal while Henry is pulling the
statue with a force of 180 newtons at a 40° angle with the horizontal.
What is the magnitude of the combined force they exert on the statue?
14. |v| = 12, θ = 42°
Find the component form of v with the given magnitude and direction angle.
11. D(-4, 3), E(5, -2) 9i − 5j
10. D(4, -5), E(6, -7) 2i − 2j
⎯⎯⎯ be the vector with the given initial and terminal points. Write DE
⎯⎯⎯
Let DE
as a linear combination of the vectors i and j.
8. v = 〈-3, 6〉
Find a unit vector u with the same direction as v.
〈-21, 28〉
7. 5w - 3v
6. 4v + 3w
〈-1, 11〉
〈-7, 7〉
5. w - 2v
〈6, -3〉
4. 3v
Find each of the following for v = 〈2, -1〉 and w = 〈-3, 5〉.
10
〈-3, -1〉; √
1. A(2, 4), B(-1, 3)
⎯⎯⎯ with the given initial and
Find the component form and magnitude of AB
terminal points.
8-2
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
48°
3. HIKING Amel is hiking in the forest.
He hikes 2 miles west and then hikes
3.4 miles north. If he would have hiked
diagonally to reach the same ending
point, how much shorter would his hike
have been? 1.5 mi
3.3i + 5.25j
2. TRANSPORTATION Jordyn is riding
the bus to school. The bus travels
north for 4.5 miles, east for 2 miles,
and then 30° north of east for 1.5 miles.
Express Jordyn’s commute as a linear
combination of unit vectors i and j.
horizontal
b. What is the resultant direction of
the javelin? 42.5° with the
a. What is the resultant speed of
the javelin? 30.8 m/s
Answers
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Chapter 8
DATE
13
PERIOD
start
end
θ
Lesson 8-2
10/23/09 11:44:15 AM
Glencoe Precalculus
c. If Walter wants to land directly in
front of his starting point, at what
angle with respect to the shore
should he kayak? 122°
b. At what angle will Walter be moving
with respect to the shore? 58°
a. What is the resultant velocity of
the kayak? 4.7 m/s
4 m/s
2.5 m/s
6. KAYAKING Walter is kayaking across
a river that has a current of 2.5 meters
per second. He is paddling at a rate of
4 meters per second perpendicular to
the shore.
427 mph
5. FLYING To reach a destination, a pilot
is plotting a course that will result in a
velocity of 450 miles per hour at an
angle of 30° north of west. The wind is
blowing 50 miles per hour to the north.
Find the direction and speed the pilot
should set to achieve the desired
resultant. 24.2° north of west,
4. AIRPLANES An airplane is traveling
300 kilometers per hour due east. A
wind is blowing 35 kilometers per hour
75° south of west. What is the resulting
speed of the airplane? 292.9 km/h
Vectors in the Coordinate Plane
Word Problem Practice
1. TRACK Monica is throwing the javelin
in a track meet. While running at
4 meters per second, she throws the
javelin with a velocity of 28 meters
per second at an angle of 48°.
8-2
NAME
Answers (Lesson 8-2)
Enrichment
PERIOD
40°
25 N
A6
-19.2 N
40°
25 N
-68.6 N
19.2 N
52.5 N
52.5
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Chapter 8
14
Glencoe Precalculus
2. What is the coefficient of friction for a person pushing a 25-kilogram box across a
carpeted floor with a force of 75 newtons if the force is being applied at 25°? 0.32
1. What is the coefficient of friction for a person pulling a 41-kilogram child on a sled if the
angle of the rope is 45° and the force of the pull is 50 newtons? 0.10
Exercises
Therefore, the coefficient of friction is 0.37.
16.1 N
≈ 19.2
x-component
≈ 16.1
y-component
Force due to gravity = mass × acceleration due to gravity
7(-9.8) = -68.6
n = 68.6 - 16.1 = 52.5
Normal force
19.2
μ=−
about 0.37
Coefficient of friction
x = 25 cos 40°
y = 25 sin 40°
A force diagram can be used to represent this situation. Calculate the x-component and the
y-component of the 25-newton force. Because the wagon is moving at a constant velocity, the
force of friction has to be equal to, but opposite the x-component. The force of gravity is
balanced by the sum of the y-component and the normal force.
Consider a person pulling a 7-kilogram wagon with
a constant velocity as shown in the diagram. The
person is pulling with a force of 25 newtons and
the wagon handle makes a 40° angle with the
horizontal. What is the coefficient of friction
for the situation?
Example
According to Newton’s first law of motion, if an object is moving with constant velocity, then
all the forces in the system are balanced. If the object is not moving at a constant velocity,
then the forces are unbalanced. So, if a person is pushing on an object and the object is
moving at a constant velocity, the force of the person pushing on the object is balanced
by the force of friction.
The coefficient of friction μ (pronounced “mu”) is the ratio of the force of friction between
two surfaces and the force pushing the surfaces together. When an object is resting on the
ground, the force pushing them together is found by multiplying the mass (in kilograms) by
acceleration due to gravity, -9.8 m/s2. The normal force n is the force of the ground
pushing back up on the object.
DATE
10/23/09 11:52:19 AM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 8
Friction and the Normal Force
8-2
NAME
DATE
Simplify and solve for θ.
-54, not orthogonal
2. u = -8i + 5j, v = 3i −6j
4 √
10
3 √
10
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005_032_PCCRMC08_893809.indd 15
Chapter 8
179.3°
5. u = 〈-3, -5〉, v = 〈7, 12〉
15
39.5°
6. u = 13i - 5j, v = 6i + 2j
Find the angle θ between u and v to the nearest tenth of a degree.
4. c = 〈−12, 4〉
3. a = 〈9, 3〉
Use the dot product to find the magnitude of the given vector.
0, orthogonal
1. u = 〈2, 4〉, v = 〈−12, 6〉
Lesson 8-3
10/23/09 11:58:37 AM
Glencoe Precalculus
Find the dot product of u and v. Then determine if u and v are orthogonal.
Exercises
The measure of the angle between u and v is about 112°.
-10 + 3
√
26 √
13
Evaluate.
θ = cos −1 − or about 112°
u = 〈5, 1〉 and v = 〈−2, 3〉
-10 + 3
cos θ = −
√
26 √
13
Angle between two vectors
〈5, 1〉 · 〈-2, 3〉
cos θ = −
|〈5, 1〉| |〈-2, 3〉|
|u| |v|
u·v
cos θ = −
and v = 〈−2, 3〉.
Find the angle θ between vectors u and v if u = 〈5, 1〉
u · v = 4(8) + 5(-6)
=2
Since u · v ≠ 0, u and v are not orthogonal.
u · v = 5(−3) + 1(15)
=0
Since u · v = 0, u and v are orthogonal.
Example 2
b. u = 〈4, 5〉, v = 〈8, -6〉
a. u = 〈5, 1〉, v = 〈-3, 15〉
Example 1 Find the dot product of u and v. Then determine if u and v
are orthogonal.
|a| |b|
PERIOD
The dot product of a = 〈a 1, a 2〉 and b = 〈b 1, b 2〉 is defined as
a · b = a 1b 1 + a 2b 2. The vectors a and b are orthogonal if and only if a · b = 0.
a·b
If θ is the angle between nonzero vectors a and b, then cos θ = −
.
Dot Products and Vector Projections
Study Guide and Intervention
Dot Product
8-3
NAME
Answers (Lesson 8-2 and Lesson 8-3)
Chapter 8
DATE
Dot Products and Vector Projections
Study Guide and Intervention
(continued)
PERIOD
〈
13 13
〉
A7
13
13 〉
〈
13
13 〉 〈 13 13 〉
9
56 72
7
7
9
, -−
; u = 〈−
, -−
+ −
,−
〈−
65 〉
65
65 〉 〈 65 65 〉
65
3. u = 〈1, 1〉, v = 〈9, -7〉
205
205
〉
16
1508 696
+ -−
,−
〈
286
132
286
132
, -−
; u = 〈- −
, -−
〈- −
205
205 〉
205 〉
205
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Glencoe Precalculus
005_032_PCCRMC08_893809.indd 16
Chapter 8
5. u = −8i + 2j, v = 6i + 13j
180 15
23
276
180 15
, − ; u = 〈−
,− + −
, -−
〈−
29 29 〉 〈 29
29 〉
29 29 〉
205
205
328
295 472
328
+ -−
, −〉
, -−
; u = 〈- −
, -−
〈- −
89
89 〉
89 〉 〈 89
89
89
2. u = 〈−7, 3〉, v = 〈8, 5〉
4. u = 7i - 9j, v = 12i + j
〈−4017 , - −1017 〉; u = 〈−4017 , - −1017 〉 + 〈−1117 , −4417 〉
1. u = 〈3, 2〉, v = 〈-4, 1〉
Find the projection of u onto v. Then write u as the sum of two orthogonal
vectors, one of which is the projection of u onto v.
Exercises
〈
13
13 〉
4 6
= 〈8, 6〉 − - −
,−
〈
108 72
= 〈−, −〉
13 13
Step 2 Find u − proj v u.
108 72
4 6
4 6
,− .
Therefore, proj v u is - −
, − and u = - −
,− + −
13
2
4 6
〈2, -3〉 or - −
,−
= -−
|〈2, -3〉|
=−
〈2, -3〉
2
〈8, 6〉 · 〈2, -3〉
( |v| )
u·v
proj v u = −
v
2
Step 1 Find the projection
of u onto v.
Example
Find the projection of u = 〈8, 6〉 onto v = 〈2, −3〉. Then write u as
the sum of two orthogonal vectors, one of which is the projection of u onto v.
( |v| )
u·v
proj v u = −
v.
2
A vector projection is the decomposition of a vector u into two
perpendicular parts, w1 and w2, in which one of the parts is parallel to another vector v.
When you find the projection of u onto v, you are finding a component of u that is parallel
to v. To find the projection of u onto v, use the formula:
Vector Projection
8-3
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Dot Products and Vector Projections
Practice
DATE
PERIOD
-11, not orthogonal
2. u = -i + 4j, v = 3i − 2j
-2, not orthogonal
3. u = 〈2, 0〉, v = 〈−1, −1〉
6 8
6 8
33
44
, − ; -−
,− + −
, -−
〈- −
25 25 〉 〈 25 25 〉 〈 25
25 〉
99
25 75
33
33
99
, -−
; −
, -−
+ −
,−
〈−
2〉 〈2
2 〉 〈2 2〉
2
32 16
12 24
12 24
, − ; -−
, − + 〈−
,−
〈- −
5 5〉
5 5〉 〈 5 5〉
Answers
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005_032_PCCRMC08_893809.indd 17
Chapter 8
17
Lesson 8-3
10/23/09 11:59:46 AM
Glencoe Precalculus
12. PHYSICS Janna is using a force of 100 pounds to push a cart up a ramp. The
ramp is 6 feet long and is at a 30° angle with the horizontal. How much work
is Janna doing in the vertical direction? (Hint: Use the sine ratio and the formula
W = F · d.) 300 ft-lb
11. TRANSPORTATION Train A and Train B depart from the same station. The path that
train A takes can be represented by 〈33, 12〉. If the path that train B takes can be
represented by 〈55, 4〉, find the angle between the pair of vectors. 15.8°
10. u = 〈−2, −1〉, v = 〈−3, 4〉
9. u = 〈62, 21〉, v = 〈−12, 4〉
8. u = 〈4, 8〉, v = 〈−1, 2〉
Find the projection of u onto v. Then write u as the sum of two orthogonal
vectors, one of which is the projection of u onto v.
7. u = 5i − 4j, v = 2i + j 65.2°
6. u = 27i + 14j, v = i − 7j 109.3°
5. u = 〈−6, −2〉, v = 〈2, 12〉 117.9°
4. u = 〈−1, 9〉, v = 〈3, 12〉 20.4°
Find the angle θ between u and v to the nearest tenth of a degree.
0, orthogonal
1. u = 〈3, 6〉, v = 〈−4, 2〉
Find the dot product of u and v. Then determine if u and v are orthogonal.
8-3
NAME
Answers (Lesson 8-3)
A8
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Chapter 8
Sample answer: As the angle
increases, the force in the
direction of the motion
decreases; so, the work done
decreases.
b. Use your results from part a to explain
why as the angle increases the work
decreases.
1532 J, 0 J
a. Determine the amount of work done
if the angle of the rope is at 40° with
the horizontal; 90° with the horizontal.
3. BOATING Shea is pulling a boat along
a dock using a rope. She exerts a force
of 200 newtons on the rope and pulls the
boat 10 meters.
2. TARGETS Two clay pigeons are thrown
at the same time. If the path of the clay
pigeons can be represented by the vectors
p = 〈42, 58〉 and c = 〈59, 73〉, what was
the measure of the angle between the clay
pigeons? 3°
13.9 units
18
PERIOD
Glencoe Precalculus
6. SCAVENGER HUNT During a
scavenger hunt, Alexis and Marty go in
different directions. If the path that
Alexis takes can be represented by
〈9, 18〉 and the path taken by Marty
can be represented by 〈-15, 12〉, who
travels the farthest distance? Alexis
60°
5. CARNIVALS A slide at a carnival has
an incline of 60°. A 50-pound girl gets
part way down the slide and stops.
Ignoring the force of friction, what is
the magnitude of the force that is
required to keep her from sliding
down farther? 43.3 pounds
1125 J
c. How much work is done by the pilot?
43.2°
b. What is the measure of the angle
made by the staircase?
11.0 m
a. What is the length of the staircase?
4. TRAVELING A pilot is carrying a bag
weighing 150 newtons up a flight of
stairs. The staircase covers a horizontal
distance of 8 meters and a vertical
distance 7.5 meters.
Dot Products and Vector Projections
Word Problem Practice
DATE
10/23/09 12:01:26 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 8
1. SUBMARINES The path of a submarine
can be described by the vector v = 〈8, 3〉.
If the submarine then changes direction
and travels along the vector u = 〈2, 5〉,
what is the distance traveled by the
submarine?
8-3
NAME
Enrichment
DATE
f(x)
-1
a + 2b + c
a
0
1
a - 2b + c
2
0
〈0, 1〉
1
〈0, -3〉
2
〈4, -11〉
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005_032_PCCRMC08_893809.indd 19
Chapter 8
19
a = 〈0, 1, –2〉 b = 〈1, –2, 0〉 c = 〈-2, 0, 1〉
4. f(x) = x 3a - xb + 3c
a = 〈1, 1, 1〉 b = 〈3, 2, 1〉 c = 〈0, 1, 2〉
3. f(x) = x 2c + 3xa - 4b
a = 〈0, 1, 1〉 b = 〈1, 0, 1〉 c = 〈1, 1, 0〉
2. f(x) = 2x 2a + 3xb - 5c
a = 〈1, 1〉 b = 〈2, 3〉 c = 〈3, –1〉
1. f(x) = x 3a - 2x 2b + 3xc
3
2
1
x
0
3
2
1
x
0
1
0
–1
x
–2
2
1
0
x
–1
f(x)
Lesson 8-3
10/23/09 12:02:42 PM
Glencoe Precalculus
〈–6, 0, 3〉
〈-7, 3, 1〉
〈-8, 12, -13〉
〈-9, 33, -51〉
f(x)
〈–12, –8, -4〉
〈-9, -4, 1〉
〈-6, 2, 10〉
〈-3, 10, 23〉
f(x)
〈–11, 3, 2〉
〈-8, -3, -1〉
〈-5, -5, 0〉
〈-2, -3, 5〉
f(x)
〈–14, –4〉
〈0, 0〉
〈6, –8〉
〈10, –22〉
For each of the following, complete the table of resulting vectors.
-1
〈4, 1〉
-2
〈12, -3〉
x
f(x)
PERIOD
a - 4b + 4c
If a = 〈0, 1〉, b = 〈1, 1〉, and c = 〈2, –2〉, the resulting vectors for the
values of x are as follows.
-2
a + 4b + 4c
x
Let a, b, and c be fixed vectors. The equation f(x) = a - 2xb + x 2c
defines a vector function of x. For the values of x shown, the
assigned vectors are given below.
Vector Equations
8-3
NAME
Answers (Lesson 8-3)
Chapter 8
DATE
Vectors in Three-Dimensional Space
Study Guide and Intervention
PERIOD
(x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2
√
(x2, y2, z2) = (7, -9, 3)
(x1, y1, z1) = (10, 2, -5) and
Distance Formula
A9
2
1
2
1
2
≈ (8.5, -3.5, -1)
0
z
(3, 2, 1)
2
y
(4, -2, -1)
x
-2
2. (4, −2, -1)
0
z
(x1, y1, z1) = (10, 2, -5) and
y
(x2, y2, z2) = (7, -9, 3)
)
( 12
5 5
2 2
19
2
)
20
Glencoe Precalculus
3/23/09 5:44:44 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Precalculus
005_032_PCCRMC08_893809.indd 20
Chapter 8
4. (-6, -12, -8), (7, -2, -11) 16.67; −, -7, - −
(
3. (8, -3, 9), (2, 8, -4) 18.06; 5, −, −
Find the length and midpoint of the segment with the given endpoints.
x
1. (3, 2, 1)
Plot each point in a three-dimensional coordinate system.
Exercises
The midpoint is at the coordinates (8.5, -3.5, -1).
1
x +x y +y z +z
10 + 7 2 + (-9) -5 + 3
, − , − ) = ( −, − , − )
(−
2
2
2
2
2
2
Use the Midpoint Formula for points in space.
b. The hikers decided to meet at the midpoint between their paths.
What are the coordinates of the midpoint?
= √(7
- 10)2 + ((-9) - 2)2 + (3 - (-5))2
≈ 13.93
The hikers are about 14 kilometers apart.
AB =
Use the Distance Formula for points in space.
a. How far apart are the hikers?
Example
HIKING The location of two hikers are represented by
the coordinates (10, 2, -5) and (7, -9, 3), where the coordinates are given
in kilometers.
Ordered triples, like ordered pairs,
can be used to represent vectors. Operations on vectors represented by ordered
triples are similar to those on vectors represented by ordered pairs.
Coordinates in Three Dimensions
8-4
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
DATE
(continued)
PERIOD
〈
√
3 √
70
3 √70
70
−, - −, - −
35
70
14
〉
⎯⎯⎯
AB = 〈6, -3, -5〉 and ⎪ ⎯⎯⎯
AB ⎥ = √
70
Unit vector in the direction of ⎯⎯⎯
AB
⎯⎯⎯
AB = 〈6, -3, -5〉
(x1, y1, z1) = (-3, 5, 1) and (x2, y2, z2) = (3, 2, −4)
Component form of vector
4 9 434
434 17 434
434
217
434
Answers
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC08_893809.indd 21
Chapter 8
3. 3x + 2y - 4z 〈19, -52, -17〉
21
Lesson 8-4
10/23/09 12:04:32 PM
Glencoe Precalculus
4. -6y + 2z 〈-10, 54, -34〉
Find each of the following for x = 3i + 2j - 5k, y = i - 5j + 7k,
and z = -2i + 12j + 4k.
3 115
115
√
√
√
, −, − 〉
〈−
115
23
√
√
√
, −, - − 〉
〈−
9 115
115
〈9, 8, 17〉, √
434 ;
2. A(-1, -4, -7), B(8, 4, 10)
〈18, -10, -6〉, 2 √
115 ;
1. A(-10, 3, 9), B(8, -7, 3)
⎯⎯⎯ with the given initial
Find the component form and magnitude of AB
⎯⎯⎯.
and terminal points. Then find a unit vector in the direction of AB
Exercises
√
70
= − or
〈6, -3, -5〉
⎪AB
⎯⎯⎯⎯⎥
⎯⎯⎯⎯
AB
u=−
Using this magnitude and component form, find a
⎯⎯⎯⎯.
unit vector u in the direction of AB
⎪AB
⎯⎯⎯⎯⎥ = √
62 + (-3)2 + (-5)2 or √
70 .
⎯⎯⎯⎯ is
Using the component form, the magnitude of AB
= 〈3 - (-3), 2 - 5, -4 - 1〉 or 〈6, -3, -5〉
⎯⎯⎯⎯ = 〈x2 - x1, y2 - y1, z2 - z1〉
AB
Example
⎯⎯⎯ with initial
Find the component form and magnitude of AB
point A(-3, 5, 1) and terminal point B(3, 2, -4). Then find a unit vector in the
⎯⎯⎯.
direction of AB
Operations on vectors represented by ordered triples are similar to
those on vectors represented by ordered pairs. Three-dimensional vectors can be added,
subtracted, and multiplied by a scalar in the same ways. In space, a vector v in standard
position with a terminal point located at (v1, v2, v3) is denoted by 〈v1, v2, v3〉. Thus, the
zero vector is 0 = 〈0, 0, 0〉 and the standard unit vectors are i = 〈1, 0, 0〉, j = 〈0, 1, 0〉,
and k = 〈0, 0, 1〉. The component form of v can be expressed as a linear combination
of these unit vectors, 〈v1, v2, v3〉 = v1i + v2 j + v3k.
Vectors in Three-Dimensional Space
Study Guide and Intervention
Vectors in Space
8-4
NAME
Answers (Lesson 8-4)
Vectors in Three-Dimensional Space
Practice
DATE
0
z
y
(-3, 4, -1)
x
(4, 7, 6)
y
x
(3, -2, 6)
4. 〈4, -2, 6〉
x
(2, 0, -5)
2. (2, 0, -5)
0
z
z
y
y
A10
〈
〉
〈
√
411
411
11 √
411 17 √
−, - −, −
411
411
411
411
〈1, -11, 17〉; √
8. A(6, 8, -5), B(7, -3, 12)
2
2
)
(
2)
7
√
445 ; -7, -6, −
10. (-17, -3, 2), (3, -9, 5)
〈-2, -4, 7〉
12. 5v - 2w
〉
Glencoe Precalculus
005_032_PCCRMC08_893809.indd 22
Chapter 8
22
Glencoe Precalculus
13. PHYSICS Suppose that the force acting on an object can be expressed by the vector
〈85, 35, 110〉, where each measure in the ordered triple represents the force in pounds.
What is the magnitude of this force? about 143 lb
〈8, -12, 14〉
11. v + w
Find each of the following for v = 〈2, -4, 5〉 and w = 〈6, -8, 9〉.
1 11
√
158 ; - −
, −, -4
(
9. (3, 4, -9), (-4, 7, 1)
Find the length and midpoint of the segment with the given endpoints.
419
419
3 √
419
17 √
11 √
−, - −, - −
419
419
419
419
〈11, -3, -17〉; √
7. A(-4, 5, 8), B(7, 2, -9)
9 √
91
91
〈−, −, - − 〉
〈- −, −, − 〉
3 √
91 √
91
91
91
91
〈3, 1, -9〉; √
17
〈-6, 4, 4〉; 2 √
3 √
17 2 √
17 2 √
17
17
17
17
6. A(4, 0, 6), B(7, 1, -3)
5. A(2, 1, 3), B(-4, 5, 7)
⎯⎯⎯⎯ with the given initial
Find the component form and magnitude of AB
⎯⎯⎯⎯.
and terminal points. Then find a unit vector in the direction of AB
z
3. 〈4, 7, 6〉
Locate and graph each vector in space.
x
1. (-3, 4, -1)
PERIOD
3/23/09 5:44:57 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 8
Plot each point in a three-dimensional coordinate system.
8-4
NAME
18 ft
3 ft
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005_032_PCCRMC08_893809.indd 23
Chapter 8
c. If the bale elevator is only 27 feet
long and the only thing that can
be changed is the 18 feet that the
farmer is away from the opening, to
the nearest foot, how close does he
need to be in order to still have two
feet past the opening? 15 ft
b. If the bale elevator needs to be 2 feet
past the opening, to the nearest foot,
how long does the bale elevator have
to be? 30 ft
a. To the nearest foot, how long should
the bale elevator be in order to reach
the opening? 28 ft
24 ft
3 ft
2. FARMING A farmer is using a bale
elevator to move bales of hay into the loft
of his barn. The opening of the loft door
is 18 feet away from where the bales will
be loaded onto the bale elevator, 3 feet to
the right of where the bales will be
loaded, and 24 feet above the ground.
The opening can be represented by the
point (18, 3, 24). The bales will be loaded
onto the elevator 3 feet above the ground.
This can be represented by (0, 0, 3).
487.35 ft)
2 ft
DATE
23
PERIOD
(1, - −12 , 6)
Lesson 8-4
10/23/09 12:05:29 PM
Glencoe Precalculus
6. BIKING A youth group is hosting a
team bike race with pairs of
competitors. Each team will switch
riders half-way through the race. If the
starting point can be represented
by (0, 0, 3) and the ending point can be
represented by the point (2, -1, 9), at
what point will the cyclists trade?
0.62 mi
5. ZIP-LINES A resort in Colorado has a
series of zip-lines that tourists can take
to travel through some wooded areas.
The platform of the first zip-line is
represented by the point (1.5, 0.5, 0.4)
and a second platform can be
represented by the point (1.8, 1, 0.2).
How long is the zip-line if the
coordinates are in miles?
-4.70i + 2.98j - 2.72k
4. ROBOTICS An underwater robot is
being used to explore parts of the ocean
floor. The robot is diving due north at
3 m/s at an angle of 65˚ with the surface
of the water. If the current is flowing at
5 m/s at an angle of 20˚ north of west,
what is the vector that represents the
resultant velocity of the underwater
robot? Let i point east, j point north,
and k point up.
b. Are they in violation of the
regulation? yes
2268 ft
a. How far apart are the planes?
3. AIRPLANE Safety regulations require
airplanes to be at least a half mile
apart when they are in the air. Two
airplanes near an airport can be
represented by the points (300, 455,
2800) and (-250, 400, 5000), where
the coordinates are given in feet.
Vectors in Three-Dimensional Space
Word Problem Practice
1. TRAVELING A family from Des Moines,
Iowa, is driving to Tampa, Florida.
According to the car’s GPS, Des Moines
is at (93.65˚, 41.53˚, 955 ft) and Tampa
is at (82.53˚, 27.97˚, 19.7 ft). Determine
the longitude, latitude, and altitude of
the halfway point. (88.09˚, 34.75˚,
8-4
NAME
Answers (Lesson 8-4)
Chapter 8
Enrichment
DATE
PERIOD
r+s-t
A11
7
7
2
24
Glencoe Precalculus
10/23/09 12:06:12 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Precalculus
005_032_PCCRMC08_893809.indd 24
Chapter 8
2
3
1
v = -−
u+−
z
3. v = 〈1, -1, 2〉, u = 〈1, 2, -1〉, w = 〈2, 2, 1〉, and z = 〈1, 0, 1〉
7
23
8
1
v= −
u-−
w+−
z
2. v = 〈5, -2, 0〉, u = 〈1, -2, 3〉, w = 〈-1, 0, 1〉, and z = 〈4, 2, -1〉
v = -5u - w + 3z
1. v = 〈-6, -2, 2〉, u = 〈1, 1, 0〉, w = 〈1, 0, 1〉, and z = 〈0, 1, 1〉
Write each vector as a linear combination of the vectors u, w, and z.
Solving the system of equations yields the solution r = 0, s = 1, and t = 2.
So, v = w + 2z.
3=r+s+t
-4 = 3r - 2s - t
-1 = r + s - t
)
r+s+t
〈-1, -4, 3〉 = r 〈1, 3, 1〉 + s 〈1, -2, 1〉 + t 〈-1, -1, 1〉 = 3r - 2s - t
(
Example
Write the vector v = 〈-1, -4, 3〉 as a linear combination of the
vectors u = 〈1, 3, 1〉, w = 〈1, -2, 1〉, and z = 〈-1, -1, 1〉.
Every vector v ∈ v3 can be written as a linear combination of any three nonparallel
vectors. The three nonparallel vectors, which must be linearly independent, are said
to form a basis for v3, which contains all vectors having 1 column and 3 rows.
The expression v = ru + sw + tz, is the sum of three vectors each multiplied by
a scalar, and is called a linear combination of the vectors u, w, and z.
Basis Vectors in Three-Dimensional Space
8-4
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Graphing Calculator Activity
DATE
PERIOD
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Answers
005_032_PCCRMC08_893809.indd 25
Chapter 8
a reflection in the xy-plane
1. Find the reflected pyramid above when
you use the transformation matrix B
below. Describe the result.
⎡1 0 0⎤
⎢
B= 0 1 0
⎢
0
-1
0
⎦
⎣
Exercises
25
See students’ work.
⎡-1 0 0⎤
B = ⎢ 0 1 0
⎢
⎣ 0 0 1⎦
Lesson 8-4
3/23/09 5:45:14 PM
Glencoe Precalculus
2. Find the transformation matrix to reflect
over the yz-plane. Check your answer by
applying it to the pyramid above.
To find the reflected image, find BA. Verify that your
answer is correct by graphing the coordinates.
To reflect the image over the xz-plane, use the
transformation matrix B.
⎡ 1 0 0⎤
⎢
B = 0 -1 0
⎢
⎣0 0 1⎦
Enter B into a graphing calculator.
Enter the vertex matrix into a graphing calculator.
A B C D E
x ⎡-2 2 2 -2 0⎤
⎢
The vertex matrix for the pyramid is y -2 -2 2 2 0 .
⎢
z ⎣-2 -2 -2 -2 2⎦
Vectors can be used to translate figures in space, and matrix multiplication can be used to
transform figures in space. A vertex matrix is a matrix whose columns are the coordinates of
the vertices of the figure with the x-coordinate represented by the first row, the y-coordinate
represented by the second row, and the z-coordinate represented by the third row.
Consider the pyramid shown at the right.
z
Use the coordinates of the vertices of the
pyramid to create a vertex matrix.
&
A(-2, -2, -2)
B(2, -2, -2)
0
"
% y
C(2, 2, -2)
#
$
D(-2, 2, -2)
x
E(0, 0, 2)
Vector Transformations with Matrices
8-4
NAME
Answers (Lesson 8-4)
PERIOD
Dot and Cross Products of Vectors in Space
Study Guide and Intervention
DATE
A12
0, orthogonal
4. u = 3i + 6j - 3k, v = -5i - 2j - 9k
2, not orthogonal
2. u = 〈-2, -4, -6〉, v = 〈-3, 7, -4〉
Glencoe Precalculus
005_032_PCCRMC08_893809.indd 26
Chapter 8
90.5°
7. u = -4i + 5j - 3k, v = -8i - 12j - 9k
80.0°
5. u = 〈5, -22, 9〉, v = 〈14, 2, 4〉
26
36.8°
Glencoe Precalculus
8. u = i + 2j - k, v = -i + 4j - 3k
41.3°
6. u = 〈4, -5, 7〉, v = 〈11, -8, 2〉
Find the angle θ between vectors u and v to the nearest tenth of a degree.
-10, not orthogonal
3. u = 〈4, -3, 8〉, v = 〈2, -2, -3〉
35; not orthogonal
1. u = 〈3, -2, 9〉, v = 〈1, 2, 4〉
Find the dot product of u and v. Then determine if u and v are orthogonal.
Exercises
The measure of the angle between u and v is about 82.3°.
89.5
Simplify and solve for θ.
Evaluate the dot product and magnitude.
12
cos θ = −
√
89 √
90
12
θ = cos -1 −
or about 82.3°
u = 〈4, 8, -3〉 and v = 〈9, -3, 0〉
〈4, 8, -3〉·〈9, -3, 0〉
cos θ = −−
⎪〈4, 8, -3〉⎥ ⎪〈9, -3, 0〉⎥
Example 2
Find the angle θ between vectors u and v if u = 〈4, 8, -3〉
and v = 〈9, −3, 0〉.
u·v
cos θ = −
Angle between two vectors
⎪u⎥ ⎪v⎥
Example 1
Find the dot product of u and v. Then determine if u and v
are orthogonal.
b. u = 〈3, −2, 1〉, v = 〈4, 5, −1〉
a. u = 〈-3, 1, 0〉, v = 〈2, 6, 4〉
u · v = u1v1 + u2v2 + u3v3
u · v = u1v1 + u2v2 + u3v3
= 3(4) + (−2)(5) + 1(−1)
= −3(2) + 1(6) + 0(4)
= 12 + (−10) -1 or 1
= −6 + 6 + 0 or 0
Since u · v ≠ 0, u and v are not orthogonal.
Since u · v = 0, u and v are orthogonal.
⎪a⎥ ⎪b⎥
10/23/09 12:06:50 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 8
Dot Products in Space The dot product of two vectors in space is an extension
of the dot product of two vectors in a plane. Similarly, the dot product of two vectors
is a scalar. The dot product of
a = 〈a1, a2, a3〉 and b = 〈b1, b2, b3〉 is defined as a · b = a1b1 + a2b2 + a3b3.
The vectors a and b are orthogonal if and only if a · b = 0.
As with vectors in a plane, if θ is the angle between nonzero vectors a and b, then
a·b
cos θ = −
.
8-5
NAME
(continued)
PERIOD
Dot and Cross Products of Vectors in Space
Study Guide and Intervention
DATE
Simplify.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC08_893809.indd 27
Chapter 8
27
Lesson 8-5
10/23/09 12:17:32 PM
Glencoe Precalculus
-8i - 28j + 12k; 〈-8, -28, 12〉 〈5, 2, 8〉 = -8(5) + (-28)(2) + 12(8) = 0;
〈-8, -28, 12〉 〈-1, 2, 4〉 = -8(-1) + (-28)(2) + 12(4) = 0
2. u = 〈5, 2, 8〉, v = 〈-1, 2, 4〉
-14i + 2j - 22k; 〈-14, 2, -22〉 〈2, 3, -1〉 = -14(2) + 2(3) + (-22)(-1)
= 0; 〈-14, 2, -22〉 〈6, -2, -4〉 = -14(6) + 2(-2) + (-22)(-4) = 0
1. u = 〈2, 3, -1〉, v = 〈6, -2, -4〉
Find the cross product of u and v. Then show that u × v is orthogonal
to both u and v.
Exercises
= 11i or 〈11, 0, 0〉
Component form
To show that u × v is orthogonal to both u and v, find the dot
product of u × v with u and u × v with v.
(u × v) · u
(u × v) · v
= 〈11, 0, 0〉 · 〈0, 4, 1〉
= 〈11, 0, 0〉 · 〈0, 1, 3〉
= 11(0) + 0(4) + 0(1)
= 11(0) + 0(1) + 0(3)
=0+0+0
=0+0+0
=0
=0
Because both dot products are zero, the vectors are orthogonal.
= 11i - 0j + 0k
⎪ ⎥ ⎪ ⎥ ⎪ ⎥
⎪ ⎥
Example
Find the cross product of u = 〈0, 4, 1〉and v = 〈0, 1, 3〉.
Then show that u × v is orthogonal to both u and v.
i j k
u×v= 0 4 1
u = 0i + 4j + k and v = 0i + j + 3k
0 1 3
0 4
0 1
4 1
=
ij+
k
Determinant of a 3 × 3 matrix
1 3
0 1
0 3
= (12 - 1)i − (0 - 0) j + (0 − 0)k
Determinants of 2 × 2 matrices
If two vectors have the same initial point and form the sides of a parallelogram, the
magnitude of the cross product will give you the area of the parallelogram.
If three vectors have the same initial point and form adjacent edges of a parallelepiped,
then the absolute value of the triple scalar product gives the volume. To find the triple
scalar product, use the same matrix set up that is used for cross products, but i, j, and
k are replaced by the third vector.
a × b = (a2b3 - a3b2)i - (a1b3 - a3b1)j + (a1b2 - a2b1)k.
If a = a1i + a2j + a3k and b = b1i + b2j + b3k, the cross product of a and b is the vector
Cross Product of Vectors in Space
Cross Products Unlike the dot product, the cross product of two vectors
is a vector. This vector does not lie in the plane of the given vectors but is
perpendicular to the plane containing the two vectors.
8-5
NAME
Answers (Lesson 8-5)
Chapter 8
PERIOD
Dot and Cross Products of Vectors in Space
Practice
DATE
3; not orthogonal
2. 〈-4, -1, 1〉 · 〈1, -3, 4〉
3. 〈0, 0, 1〉 · 〈1, -2, 0〉
0; orthogonal
about 96.9°
5. u = 〈3, -2, 1〉,
v = 〈-4, -2, 5〉
about 51.3°
6. u = 〈2, -4, 4〉,
v = 〈-2, -1, 6〉
A13
〈1, 4, -7〉; 〈1, 4, -7〉 · 〈4, -1, 0〉
= (1)(4) + (4) (-1) + (-7)(0) = 0;
〈1, 4, -7〉 · 〈5, -3, -1〉 = (1)(5)
+ (4)(-3) + (-7)(-1) = 0
74.2 units2
12. u = 〈2, 0, -8〉, v = 〈-3, -8, -5〉
28
Glencoe Precalculus
10/23/09 12:18:59 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Precalculus
005_032_PCCRMC08_893809.indd 28
Chapter 8
8.2 newton meters
14. TOOLS A mechanic applies a force of 35 newtons straight down to a ratchet
that is 0.25 meter long. What is the magnitude of the torque when the handle
makes a 20° angle above the horizontal?
643 units3
13. Find the volume of the parallelepiped with adjacent edges represented by the
vectors 〈3, -2, 9〉, 〈6, -2, -7〉, and 〈-8, -5, -2〉.
62.4 units2
11. u = 〈9, 4, 2〉, v = 〈6, -4, 2〉
Find the area of the parallelogram with adjacent sides u and v.
〈7, 1, -11〉; 〈7, 1, -11〉 · 〈3, 1, 2〉
= (7)(3) + (1)(1) + (-11)(2) = 0;
〈7, 1, -11〉 · 〈2, -3, 1〉 = (7)(2)
+ (1)(-3) + (-11)(1) = 0
10. 〈4, -1, 0〉 × 〈5, -3, -1〉
= (27)(3) + 3(1) + (14)(-6) = 0;
〈27, 3, 14〉 · 〈-2, 4, 3〉
= (27)(-2) + (3)(4) + (14)(3) = 0
9. 〈3, 1, 2〉 × 〈2, -3, 1〉
〈27, 3, 14〉; 〈27, 3, 14〉 · 〈3, 1, -6〉
= -3(1) + (-3)(3) + (3)(4) = 0;
〈-3, -3, 3〉 · 〈-1, 0, -1〉
= (-3)(-1) + (-3)(0) + 3(-1) = 0
8. 〈3, 1, -6〉 × 〈-2, 4, 3〉
〈-3, -3, 3〉; 〈-3, -3, 3〉 · 〈1, 3, 4〉
7. 〈1, 3, 4〉 × 〈-1, 0, -1〉
Find the cross product of u and v. Then show that u × v is orthogonal
to both u and v.
about 154.9°
4. u = 〈1, -2, 1〉,
v = 〈0, 3, -2〉
Find the angle θ between vectors u and v to the nearest tenth of a degree.
-9; not orthogonal
1. 〈-2, 0, 1〉 · 〈3, 2, -3〉
Find the dot product of u and v. Then determine if u and v are orthogonal.
8-5
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
v = 〈-9, 13, 12〉. What is the surface
area of the sculpture? 1802 units2
4. SCULPTURE A parallelepiped sculpture
is being created. When the sculpture is
set, three adjacent edges can be
represented by the vectors
t = 〈15, 12, 10〉, u = 〈13, -8, -5〉, and
3. MIRROR Two adjacent edges of a mirror
in a dressing room are represented by
the vectors 〈3, 4, 2〉 and 〈-4, 4, 3〉. What
is the area of the mirror? 33 units2
from the axis of rotation as
possible
2. REVOLVING DOOR Erica is standing
in a revolving door that is not moving.
If Erica wants to produce just enough
torque to make the door rotate but wants
to apply the least amount of force, where
should she push on the door with respect
to the axis of rotation? as far away
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC08_893809.indd 29
Answers
Chapter 8
DATE
PERIOD
29
〈9, -21, 30〉
Lesson 8-5
10/29/09 12:30:46 PM
Glencoe Precalculus
d. If the velocity of the rockets
remains constant, what vectors
would represent the rockets at
3 seconds? 〈9, 18, 36〉,
c. What is the measure of the angle
between the two rockets? 59.8°
b. What vector represents the path of
the second rocket? 〈3, -7, 10〉
a. What vector represents the path of
the first rocket? 〈3, 6, 12〉
6. ROCKETS Two rockets are launched
simultaneously. The first rocket starts
at the point (0, 1, 0) and after 1 second
is at the point (3, 7, 12). The second
rocket starts at the point (0, -1, 0)
and after 1 second is at the
point (3, -8, 10).
60° 9
5. BICYCLING A cyclist applies a force
straight down on a bicycle pedal, as
shown in the diagram. The length to
the pedal’s axle is 0.2 meter and the
angle created with the vertical is 60°.
The magnitude of the torque is 150
newton meters. Find the force applied to
the pedal. 866 N
Dot and Cross Products of Vectors in Space
Word Problem Practice
1. MECHANIC A mechanic is setting the
timing of an engine. He is using a ratchet
to turn the crankshaft. The ratchet is
0.5 meter long and the mechanic applies
22 newtons of force straight down on the
handle when the handle is at a 25° angle
with the horizontal. What is the
magnitude of the torque? 10.0 N-m
8-5
NAME
Answers (Lesson 8-5)
Enrichment
PERIOD
A14
2
Glencoe Precalculus
005_032_PCCRMC08_893809.indd 30
Chapter 8
no
5. 〈2, -4, 6〉, 〈3, -1, 2〉, 〈-6, 8, 10〉
no
3. 〈1, 1, 1〉, 〈-1, 0, 1〉, 〈1, -1, -1〉
yes; Sample answer:
〈-2, 6〉 + 2〈1, -3〉 = 〈0, 0〉
1. 〈-2, 6〉, 〈1, -3〉
30
no
4
〉
Glencoe Precalculus
9
6. 〈1, -2, 0〉, 〈2, 0, 3〉, -1, 1, −
〈
yes; Sample answer: 2〈1, 1, 1〉
- 〈-1, 0, 1〉 + 〈-3, -2, -1〉
= 〈0, 0, 0〉
4. 〈1, 1, 1〉, 〈-1, 0, 1〉, 〈-3, -2, -1〉
yes; Sample answer:
2〈3, 6〉 - 3〈2, 4〉 = 〈0, 0〉
2. 〈3, 6〉, 〈2, 4〉
Determine whether the given vectors are linearly dependent. Write yes or
no. If yes, give a linear combination that yields a zero vector.
dependent.
〈-1, 2, 1〉 + 〈1, -1, 2〉 + −1 〈0, -2, -6〉 = 〈0, 0, 0〉, so the three vectors are linearly
2
1
Hence, one solution is a = 1, b = 1, and c = −
.
The above system does not have a unique solution. Any solution must satisfy
the conditions that a = b = 2c.
-a + b
=0
2a - b - 2c = 0
a + 2b - 6c = 0
Solve a〈-1, 2, 1〉 + b〈1, -1, 2〉 + c〈0, -2, -6〉 = 〈0, 0, 0〉.
Example
Are the vectors 〈-1, 2, 1〉, 〈1, -1, 2〉, and 〈0, -2, -6〉
linearly dependent?
The zero vector is 〈0, 0〉 in two dimensions and 〈0, 0, 0〉 in three dimensions.
A set of vectors is called linearly dependent if and only if there exist scalars,
not all zero, such that a linear combination of the vectors yields a zero vector.
DATE
3/23/09 5:45:43 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Chapter 8
Linearly Dependent Vectors
8-5
NAME
Graphing Calculator Activity
DATE
PERIOD
2nd
ENTER
4
9
3
ENTER
ENTER
[QUIT]
ENTER
(–)
1
ENTER
3
(–)
ENTER
ENTER
3
ENTER
ENTER
1
(–)
(–)
5
ENTER
[MATRIX]
ENTER
2nd
[MATRIX]
ENTER
)
ENTER
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
005_032_PCCRMC08_893809.indd 31
Chapter 8
31
Lesson 8-5
3/23/09 5:45:46 PM
Glencoe Precalculus
3. The volume of a parallelepiped is 112 cubic units. Three adjacent sides are
t = 〈-10, 3, 4〉, u = 〈-8, 7, 3〉, and v = 〈-6, -2, x〉. Is 3, 5, or 9 the correct
value of x? 5
u = 〈-31, 3, 22〉, and v = 〈-65, 31, 0〉. 42,936 units3
2. Find the volume of the parallelepiped with adjacent sides t = 〈8, -22, 90〉,
u = 〈4, 3, 9〉, and v = 〈2, 3, 1〉. 366 units3
1. Find the volume of the parallelepiped with adjacent sides t = 〈6, -12, -9〉,
Exercises
The determinant is -350. Volume cannot be negative, so the volume is
350 cubic units.
2nd
Now use the MATH menu to calculate the following determinant.
ENTER
7
(–)
8
[MATRIX]
ENTER
2
2nd
Enter the data into the graphing calculator under matrix A.
Example
Find the volume of a parallelepiped with adjacent sides
t = 〈3, -2, 8〉, u = 〈4, 9, -1〉, and v = 〈-1, -5, -7〉.
The volume of a parallelepiped with adjacent sides t = 〈t1, t2, t3〉, u = 〈u1, u2, u3〉,
and v = 〈v1, v2, v3〉 can be calculated by finding the determinant of the
matrix below.
⎡t1 t2 t3 ⎤
⎢u u u ⎢ 1 2 3
⎣v1 v2 v3 ⎦
Use Matrices to Find the Volume of Parallelepipeds
8-5
NAME
Answers (Lesson 8-5)
Chapter 8 Assessment Answer Key
(Lessons 8-1 and 8-2)
Page 33
A
2.
5.5 cm; 29°
3.
4.
(Lesson 8-4)
Page 34
12.93 mm,
1.36 mm
1.
Quiz 3
1.
2.
3.
Page 35
〈5, 0, -12〉; 13
〈-7, -5〉
-15i + 5j
6.
D
2.
H
3.
A
4.
H
5.
A
11
, 10〉
〈3, - −
3
D
4.
5.
1.
〈3, 11, 10〉
5.9 cm; 187°
5.
Mid-Chapter Test
22.7
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Quiz 4 (Lesson 8-5)
8b.
254.5 lb
〈-15, 20〉
〈21, -24〉
8c.
〈6, -10〉
7.
8a.
Quiz 2 (Lesson 8-3)
Page 33
1.
Answers
Quiz 1
〈
76 57
25 25
〈− −〉
6.
〈 〉
u=
1.
〉
−, − ;
Page 34
9; not
orthogonal
〈18, 38, 25〉;
〈18, 38, 25〉 ·
〈5, -5, 4〉 = 0;
〈18, 38, 25〉 ·
2. 〈2, 3, -6〉 = 0
149 168
,
25
25
+
76 57
25 25
−, −
7.
86.2°
19.2 N, 16.1 N
3.
84.8°
8.
3.
A
54.2°
4.
36 54
,−
〈- −
13 13 〉
2.
4.
7.9 newton
meters
9a.
5.
Chapter 8
536.2 J
5.
D
9b.
A15
〈3.8, 0〉,
〈6.2, 3.3〉
10.5 m/s
at 18.2°
Glencoe Precalculus
Chapter 8 Assessment Answer Key
Vocabulary Test
Page 36
Form 1
Page 37
1.
A
2.
H
Page 38
11.
D
12.
J
13.
A
14.
H
15.
A
16.
G
17.
D
18.
J
19.
D
20.
H
1. component form
2. initial point
3.
B
3. orthogonal
4. magnitude
4.
5. parallel vectors
5.
A
7. unit vector
6.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6. ordered triple
F
G
8. Opposite vectors
9. terminal point
10. standard position
11. Sample answer:
a directional
measurement
where the angle is
measured clockwise
from north
12. Sample answer:
the sum of two or
more vectors
Chapter 8
7.
C
8.
J
9.
D
10.
H
B:
A16
〈36, -22, -16〉
Glencoe Precalculus
Chapter 8 Assessment Answer Key
Page 40
11.
1.
B
2.
H
Form 2B
Page 41
12.
C
H
1.
B
2.
J
13. B
3.
C
3.
14.
F
4.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4.
5.
A
F
13.
D
14.
G
15.
C
C
16.
J
6.
H
B
8. H
9.
12.
J
B
10. G
Chapter 8
18.
19.
J
7.
D
8.
G
A
9.
10.
A17
17.
A
18.
H
19.
A
20.
H
D
20. H
45
57
27
, -−
, -−
−
〈
2
8
2〉
B:
H
H
17. A
7.
B
F
16.
6.
11.
A
15. C
5.
Page 42
H
B: 〈270, 162, -180〉
Glencoe Precalculus
Answers
Form 2A
Page 39
Chapter 8 Assessment Answer Key
Form 2C
Page 43
1.
2.
Page 44
9.99 m, 0.35 m
〈
34
3 √
34 5 √
34
34
〉
- −, −
14.
15.
3.
6.0 cm, 219˚
4.
116.6˚
5.
〈-1, 15〉; 15.03
6.
17.5N; 63.8˚
south of east
16.
〈35, 52〉
17.
85
17
, -−
〈−
26
26 〉
18.
536.5 units2
19.
16.1 miles
183.2 J
〈18, 48, 28〉
145.4°
11.
20.
12.
13.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
〈28, -16, 15〉
9.
10.
660.2 mph
〈7, -21〉
7.
8.
38.8N, 48.9°
76.5 units3
−12, not
orthogonal
6, not
orthogonal
Chapter 8
B:
A18
180°
Glencoe Precalculus
Chapter 8 Assessment Answer Key
Form 2D
Page 45
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2.
4.33 m, 2.5 m
〈
〉
-3 √
73 8 √
73
−, −
73
73
3.
2.2 cm, 43˚
4.
53.1˚
5.
〈-4, -1〉; 4.12
6.
20.4 N; 80.35˚
south of east
7.
14.
29.6 N, 58.4°
15.
616.4 mi/h
16.
〈30, 21〉
17.
111
74
, -−
〈−
13
13 〉
18.
655 units2
〈-3, 11〉
8.
〈1, -10, 7〉
9.
213.7 J
19.
10.
〈5, 13, 1〉
11.
97.7˚
0, orthogonal
−4,
13. not orthogonal
Answers
1.
Page 46
20.4 miles
20.
113.5 units3
B:
0˚
12.
Chapter 8
A19
Glencoe Precalculus
Chapter 8 Assessment Answer Key
Form 3
Page 47
Page 48
1. 10.57 m, 4.27 m
〈
√
10
3 √
10
-− , -−
10
10
2.
〉
0, orthogonal
−8,
not
orthogonal
13.
12.
14.
3.
4.2 cm, 43˚
4.
4.8˚
5.
306.0 mi/h;
about 52.5°
15. north of east
〈-1.9, 0〉; 1.9
20.4 N, 3.7˚
south of east
16.
〈-2, -19〉
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
6.
26.6 N, 51.6°
17. 〈-0.87, -0.61〉
7.
〈−23 , - −34 〉
8.
81
193
, -−
, 4〉
〈−
2
8
18.
19.
250.3 units2
plane 2; 0.52 mi
607.1 N, 19.3˚
9. north of east
10.
35
, 14, 14〉
〈- −
6
11.
57.4°
20.
B:
Chapter 8
A20
98.1 units3
180˚
Glencoe Precalculus
Chapter 8 Assessment Answer Key
Page 49, Extended-Response Test
Sample Answers
1a.
⎢i j k⎢
1h. 2 1 0 = 0i + 0j + 5k
⎢1 3 0 ⎢
B+C
⎢
C
B
B
C
⎢
B+C
1b. a - b = a + (-b ), as shown in the
figure below.
2a. 12.12 units
-C
2b. 5.39 units
B-C
B
1c. Yes. They are the same diagonal of a
parallelogram.
2c. second plane
B
B+C
3. Sample answer: The vectors a = 〈1, 0〉
and b = 〈0, -1〉 are perpendicular
because their dot product is
a1b1 + a2b2 = 1(0) + 0(-1) or 0;
a = 〈5, 5〉 and b = 〈5, -5〉 are
perpendicular because their dot
product is a1b1 + a2b2 = 5(5) + 5(-5) =
25 - 25 = 0.
C+B
C
B
1d. No. a - b and b - a are shown in the
figures below.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
C
C− B
−C
−B
B−C
B
1e. Add the first terms of each vector
together, and then add the second terms
together. These terms represent the
horizontal and vertical components of
the resultant vector, respectively.
c + d = 〈-3 + -8, 1 + (-11)〉, or
〈-11, -10〉
The magnitude of
c + d is √
(-11)2 + (-10)2 , or about 14.9.
1f. Sample answer: 〈1, 2, 3〉 - 〈-3, 3, 0〉
= 〈4, -1, 3〉; 〈4, -1, 3〉 = 4i - j + 3k
1g. Sample answer: 〈3, 7〉; The vectors are
perpendicular because their dot product
is zero.
a1b1 + a2b2 = 7 · 3 + (-3)7 = 0
Chapter 8
A21
Glencoe Precalculus
Answers
C
Chapter 8 Assessment Answer Key
Standardized Test Practice
Page 50
1.
A
B
C
D
2.
F
G
H
J
3.
4.
A
F
B
G
C
H
11.
A
B
C
D
12.
F
G
H
J
13.
A
B
C
D
14.
F
G
H
J
15.
A
B
C
D
16.
F
G
H
J
17.
A
B
C
D
18.
F
G
H
J
19.
A
B
C
D
20.
F
G
H
J
D
J
A
B
C
D
6.
F
G
H
J
7.
A
B
C
D
8.
F
G
H
J
9.
A
B
C
D
10.
F
G
H
J
A22
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5.
Chapter 8
Page 51
Glencoe Precalculus
Chapter 8 Assessment Answer Key
Standardized Test Practice
(continued)
Page 52
f(x) = 3 sin (6x) + 9
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
22.
x=5
23.
11.5 m/s, 32°
24.
x = −2, y = 5
25.
66,221 J
26.
73.3°
27.
CB = 17.8,
A = 52.2°,
B = 37.8°,
C = 90°
28.
5.4 m/s, 36.2°
Answers
21.
(x – 9)2 + (y + 2)2
= 25
29.
30a.
〈45, 18, -1.5〉
30b.
〈-13, 28, -2〉
30c. Plane 2, 17.6 miles
Chapter 8
A23
Glencoe Precalculus