Solution for Linear Algebra Homework: Vectors and Quaternions Problems

This document provides a detailed solution to a linear algebra homework assignment, focusing on vector operations and quaternion representations. The solution begins with calculating the scalar multiple of a vector to satisfy the dot product condition. It then proceeds to calculate the projection of one vector onto another, followed by the reflection of a vector in a plane. The assignment also delves into quaternions, including the calculation of a unit quaternion, and rotations represented by quaternions. The solution provides step-by-step calculations and explanations for each problem, including quaternion rotations about the x and z axes, and the composite quaternion representing the combined rotations, helping students understand and solve similar linear algebra problems.

Solution
Q8)
a)
u∗v =0
u=(a ,b ,c )
If the vector matrix is scales by 2
u 1=2 u=2(a , b , c)
¿( 2a ,2 b , 2 c)
v=(0.5 ,−0.1 , 0.9)
u∗v =[0.5 ( 2 a )−0.1 ( 2 b ) +0.9 ( 2 c ) ]=0
Let
a = 2
b= 1
substituting
[0.5 ( 2∗2 )−0.1 ( 2∗1 )+ 0.9 ( 2c ) ]=0
2-0.2 + 1.8c = 0
1.8c = -1.8
C = -1.0
To confirm
[0.5 ( 2∗2 )−0.1 ( 2∗1 )+ 0.9 ( 2∗−1 ) ]=0
u=(2 , 1 ,−1)
b)
Projuv = (2, 1, -1)(0.5, -0.1, 0.9)
= (2*0.5, -0.1*1, -1*0.9)
= (1, -0.1, -0.9)

c)
a= ( 0.5 ,−0.1 , 0.9)
√0.52+−0.12+ 0.92
r =u+ v = (2, 1, -1) + (0.5, -0.1, 0.9)
= (2.5, 0.9, -0.1)
r' obtained by reflection of r∈the plane
r '=r−2 (a . r)a
= (2.5, 0.9, -0.1) - 2 [ ( 0.5 ,−0.1 , 0.9 )
1.0344 ∗( 2.5,0 .9 ,−0.1 ) ]∗[ ( 0.5 ,−0.1, 0.9 )
1.0344 ]
= (2.5, 0.9, -0.1) – 1.869187947(0.625, 0.009, -0.081)
= (1.331, 0.883, 0.0515)
Q9)
q = ¿
θ=700∧v=(0 , 3 , 4)
||v|| = √02 +32 +42=5
The unit quaternion, q = [ √3+ √9
3 ,
√3− √9
3 ∗(0 , 3 , 4)
5 ]
= [ √6
3 , 0 ,0 ]
Q10) Answer the following questions
a) What is the quaternion q1 that represents the rotation of 270 degree about the x-axis
The fixed axis is x and angle of rotation θ=2700

That is, u=
[1
0
0 ]
Sin(270/2)u= 1
√2 [1
0
0 ]
Sin(135) = 1
√2 [1
0
0 ]
Cos(270/2) = cos(135) = −1
√2 [1
0
0 ]Therefore,
q1 = [ −1
√ 2 , 1
√ 2 , 0 , 0 ]
b) What is the quaternion q2 that represents the rotation of 270 degree about the z-axis
The fixed axis is z and angle of rotation θ=2700
That is, u=
[0
0
1 ]
Sin(-270/2) u=−1
√ 2 [ 1
0
0 ]
Sin(135) = −1
√2 [0
0
1 ]
Cos(270/2) = cos(135) = −1
√2 [0
0
1 ]Therefore,
q2 = [−1
√2 , 0 ,0 ,− 1
√2 , ]
c) What rotation does the composite quaternion q = q1q2 represents? Write your answer with
specifying the rotation angel and axis of this composite quaternion

q = [ −1
√ 2 , 1
√ 2 , 0 , 0 ]∗
[ −1
√ 2 , 0 , 0 ,− 1
√ 2 , ]
q = [ 1
2 ,0 ,0 , 0 ]
The angle of rotation 300 about y-axis