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Mathematic Methods HW Solutions 15

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

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Chapter 3 15 10.3 (a) Label the vectors A , B , C , D . Then cos( A , B ) = 1 15 , cos( A , C ) = 2 3 , cos( A , D ) = 3 23 , cos( B , C ) = 2 3 15 , cos( B , D ) = radicalBig 17 690 , cos( C , D ) = 21 6 23 . (b) (1 , 0 , 0 , 5 , 0 , 1) and (0 , 0 , 1 , 0 , - 3 , 0) 10.4 (a) e 1 = (0 , 1 , 0 , 0), e 2 = (1 , 0 , 0 , 0), e 3 = (0 , 0 , 3 , 4) / 5 (b) e 1 = (0 , 0 , 0 , 1), e 2 = (1 , 0 , 0 , 0), e 3 = (0 , 1 , 1 , 0) / 2 (c) e 1 = (1 , 0 , 0 , 0), e 2 = (0 , 0 , 1 , 0), e 3 = (0 , 1 , 0 , 2) / 5 10.5 (a) bardbl A bardbl = 43, bardbl B bardbl = 41, | Inner product of A and B | = 74 (b) bardbl A bardbl = 7, bardbl B bardbl = 60, | Inner product of A and B | = 5 11.5 θ = 1 . 1 = 63 . 4 11.11 parenleftbigg x y parenrightbigg = 1 5 parenleftbigg 1 3 - 1 2 parenrightbiggparenleftbigg x prime y prime parenrightbigg , not orthogonal In the following answers, for each eigenvalue, the components of a corresponding eigenvector are listed in parentheses. 11.12 4 (1 , 1) - 1 (3 , - 2) 11.13 3 (2 , 1) - 2 ( - 1 , 2) 11.14 4 (2 , - 1) - 1 (1 , 2) 11.15 1 (0 , 0 , 1) - 1 (1 , - 1 , 0) 5 (1 , 1 , 0) 11.16 2 (0 , 1 , 0) 3 (2 , 0 , 1) - 2 (1 , 0 , - 2) 11.17 7 (1 , 0 , 1) 3 (1 , 0 , - 1) 3 (0 , 1 , 0) 11.18 4 (2 , 1 , 3) 2 (0 , - 3 , 1) - 3 (5 , - 1 , - 3) 11.19 3 (0 , 1 , - 1) 5 (1 , 1 , 1) - 1 (2 , - 1 , - 1) 11.20 3 (0 , - 1 , 2) 4 (1 , 2 , 1) - 2 ( - 5 , 2 , 1) 11.21 - 1 ( - 1 , 1 , 1) 2 (2 , 1 , 1) - 2 (0 , - 1 , 1) 11.22 - 4 ( - 4 , 1 , 1) 5 (1 , 2 , 2) - 2 (0 , - 1 , 1) 11.23
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