Mathematic Methods HW Solutions 15

Mathematic Methods HW Solutions 15 - Chapter 3 15 10.3 (a)...

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Unformatted text preview: Chapter 3 15 10.3 (a) Label the vectors A, B, C, D. Then cos(A, B) = cos(A, C) = √ 2 √3 , cos(B, C) 3 , cos(A, D) = 23 √ 17 , cos(C, D) = 6√21 . 690 23 = √1 , 15 2 √, 3 15 cos(B, D) = (b) (1, 0, 0, 5, 0, 1) and (0, 0, 1, 0, −3, 0) 10.4 (a) e1 = (0, 1, 0, 0), e2 = (1, 0, 0, 0), e3 = (0, 0, 3, 4)/5 √ (b) e1 = (0, 0, 0, 1), e2 = (1, 0, 0, 0), e3 = (0, 1, 1, 0)/√ 2 (c) e1 = (1, 0, 0, 0), e2 =√ , 0, 1, 0), e3 = (0, 1, 0, 2)/ 5 (0 √ √ 10.5 (a) A = 43, B √ 41, |Inner product of A and B| √ 74 = = (b) A = 7, B = 60, |Inner product of A and B| = 5 11.5 θ = 1.1 = 63.4◦ 1 x 13 = 11.11 y −1 2 5 x y , not orthogonal In the following answers, for each eigenvalue, the components of a corresponding eigenvector are listed in parentheses. 11.12 4 (1, 1) 11.13 3 (2, 1) −1 (3, −2) −2 (−1, 2) 11.14 4 (2, −1) 11.15 1 (0, 0, 1) −1 (1, 2) −1 (1, −1, 0) 5 (1, 1, 0) 11.16 2 (0, 1, 0) 11.17 7 (1, 0, 1) 3 (2, 0, 1) 3 (1, 0, −1) −2 (1, 0, −2) 3 (0, 1, 0) 11.18 4 (2, 1, 3) 11.19 3 (0, 1, −1) 2 (0, −3, 1) 5 (1, 1, 1) −3 (5, −1, −3) −1 (2, −1, −1) 11.20 3 (0, −1, 2) 11.21 −1 (−1, 1, 1) 4 (1, 2, 1) 2 (2, 1, 1) −2 (−5, 2, 1) −2 (0, −1, 1) 11.22 −4 (−4, 1, 1) 5 (1, 2, 2) −2 (0, −1, 1) 11.23 18 (2,2,-1) Any two vectors orthogonal to (2,2,-1) and to each 9 9 other, for example : (1,-1,0) and (1,1,4) 11.24 11.25 8 −1 −1 1 2 −2 11.26 4 1 1 11.27 D = 11.28 D = (2,1,2) Any two vectors orthogonal to other, for example : (1,0,-1) and (−1, 1, 1) (1, 1, 0) (1, −1, 2) (1,1,1) Any two vectors orthogonal to other, for example : (1,-1,0) and 1 30 11 , C= √ 01 2 −1 1 1 12 10 , C= √ −2 1 06 5 (2,1,2) and to each (1,-4,1) (1,1,1) and to each (1,1,-2) ...
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This note was uploaded on 02/29/2012 for the course MHF 2312 taught by Professor Dr.chet during the Fall '11 term at UNF.

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