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Mathematic Methods HW Solutions 14

Mathematic Methods HW Solutions 14 - Chapter 3 14 10 0 0 0...

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Chapter 3 14 7.30 R = 0 - 1 0 1 0 0 0 0 1 , S = 1 0 0 0 0 - 1 0 1 0 ; R is a 90 rotation about the z axis; S is a 90 rotation about the x axis. 7.31 From problem 30, RS = 0 0 1 1 0 0 0 1 0 , SR = 0 - 1 0 0 0 - 1 1 0 0 ; RS is a 120 rotation about i + j + k ; SR is a 120 rotation about i - j + k . 7.32 180 rotation about i - k 7.33 120 rotation about i - j - k 7.34 Reflection through the plane y + z = 0 7.35 Reflection through the ( x,y ) plane, and 90 rotation about the z axis. 8.1 In terms of basis u = 1 9 (9 , 0 , 7), v = 1 9 (0 , - 9 , 13), the vectors are: u - 4 v , 5 u - 2 v , 2 u + v , 3 u + 6 v . 8.2 In terms of basis u = 1 3 (3 , 0 , 5), v = 1 3 (0 , 3 , - 2), the vectors are: u - 2 v , u + v , - 2 u + v , 3 u . 8.3 Basis i , j , k . 8.4 Basis i , j , k . 8.6 V = 3 A - B 8.7 V = 3 2 (1 , - 4) + 1 2 (5 , 2) 8.17 x = 0, y = 3 2 z 8.18 x = - 3 y , z = 2 y 8.19 x = y = z = w = 0 8.20 x = - z , y = z 8.21 x 1 y 1 z 1 1 x 2 y 2 z 2 1 x 3 y 3 z 3 1 x 4 y 4 z 4 1 = 0 8.22 a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 = 0 8.23 For λ = 3, x = 2 y ; for λ = 8, y = - 2 x 8.24 For λ = 7, x = 3 y ; for λ = - 3, y = - 3 x 8.25 For λ = 2: x = 0, y = - 3 z ; for λ = - 3: x = - 5 y , z = 3 y ; for λ = 4: z = 3 y , x = 2 y . 8.26 r = (3 , 1 , 0) + ( - 1 , 1 , 1) z 8.27 r = (0 , 1 , 2) + (1 , 1 , 0) x 8.28 r = (3 , 1 , 0) + (2 , 1 , 1) z 9.3 A = 1 2 i 1 0 2 1 - i - 5 i 0 0 , A - 1 = 1 10
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