v1 := 80, 4<
u1 := 8Sqrt@3D, 1<
u1.v1
4
Graphics@8Thick, Blue, Arrow@880, 0<, 80, 4<<D, Red, Arrow@880, 0<, 8Sqrt@3D, 1<<D<D
v2 := 8-1  3, 2  3<
u2 := 82, -4<
u2.v2
10
3
2
MTH2301Lab6.nb
Graphics@8Thick, Blue, Arrow@880, 0<, 8-1  3, 2  3<<D, Red, Arrow@880, 0<, 82, -4<<D<D
u3 := 81, -2<
v3 := 81, 3<
pro31 = Projection@u3, v3D
:-
1
3
,-
2
2
>
pro32 = Hu3.v3L  Norm@v3D ^ 2 v3
:-
1
3
,-
2
2
>
MTH2301Lab6.nb
Graphics@8Thick, Blue, Arrow@880, 0<, 81, -2<<D, Red,
Arrow@880, 0<, 81, 3<<D, Green, Arrow@880, 0<, 8-1  2, -3  2<<D,
Dashed, Orange, Arrow@880, 0<, 83  2, -1  2<<D<D
3
4
MTH2301Lab6.nb
u4 := 8Sqrt@3D, 1<
v4 := 8-Sqrt@3D, -1<
pro41 = Projection@u4, v4D
: 3 , 1>
proj42 = Hu4.v4L  v4.v4 * v4
: 3 , 1>
Graphics@8Thick, Blue, Arrow@880, 0<, 8Sqrt@3D, 1<<D, Red,
Arrow@880, 0<, 8-Sqrt@3D, -1<<D, Dashed, Green, Arrow@880, 0<, 8Sqrt@3D, 1<<D<D
u5 := 81, 1, 1<
v5 := 82, 1, -1<
VectorAngle@u5, v5D
2
theta51 = ArcCosB
3
2
ArcCosB
3
F
F
Theta52 = ArcCos@u5.v5  HNorm@u5D Norm@v5DLD
2
ArcCosB
3
F
u6 := 82, -3, 1<
v6 := 81, -2, 1<
theta61 = VectorAngle@u6, v6D
3
7
3
ArcCosB
2
F
MTH2301Lab6.nb
theta62 = ArcCos@u6.v6  HNorm@u6D Norm@v6DLD
3
7
3
ArcCosB
F
2
Graphics3D@8Thick, Blue, Line@880, 0, 0<, 82, -3, 1<<D, Red,
Line@880, 0, 0<, 81, -2, 1<<D<D
u7 := 82, 1, 2<
v7 := 80, 3, 4<
proj71 = Projection@u7, v7D
:0,
33
44
,
25
25
>
proj72 = Hu7.v7L  Norm@v7D ^ 2 v7
:0,
33
44
,
25
25
>
u8 := 83 a, -a, 5 a<
Alp = Assuming@a > 0, ArcCos@3 a  Norm@u8DDD
F
3
ArcCosB
35
Bet = Assuming@a > 0, ArcCos@-a  Norm@u8DDD
1
ArcCosB35
F
5
6
MTH2301Lab6.nb
Gam = Assuming@a > 0, ArcCos@5 a  Norm@u8DDD
5
ArcCosB
7
F
Cos@AlpD ^ 2 + Cos@BetD ^ 2 + Cos@GamD ^ 2
1
Alp = N@Assuming@a > 0, ArcCos@3 a  Norm@u8DDDD
1.03899
Gam = N@Assuming@a > 0, ArcCos@5 a  Norm@u8DDDD
0.563943
Gam = N@Assuming@a > 0, ArcCos@5 a  Norm@u8DDDD
0.563943