sol51 - Partial Solution Set Leon Section 5.1 5.1.1c Find...

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Partial Solution Set, Leon Section 5.1 5.1.1c Find the angle between v = (4 , 1) T and w = (3 , 2) T . Solution : We have v T w = 14, || v || = 17, and || w || = 13, so the angle between v and w is θ = arccos 14 221 . 5.1.2c Using the same vectors as in the preceding problem, the vector projection of v onto w is 14 13 w = (42 / 13 , 28 / 13) T . The vector projection of w onto v is 14 17 v = (56 / 17 , 14 / 17) T . 5.1.3 For each pair of vectors x and y , compute the vector projection p of x onto y , and verify that p is orthogonal to x - p . (b) x = (3 , 5) T , y = (1 , 1) T (d) x = (2 , - 5 , 4) T , y = (1 , 2 , - 1) T . Solution : (b) The vector projection is p = x T y y T y y = 4 y = (4 , 4) T . Clearly p is orthogonal to x - p = ( - 1 , 1) T since their dot product is zero. (d) The vector projection is p = x T y y T y y = - 2 y = ( - 2 , - 4 , 2) T . As in (b), the verifica- tion of orthogonality is trivial. 5.1.5 Find the point on the line y = 2 x that is closest to the point (5 , 2). Solution : We want to find the vector projection of v = (5 , 2) T onto some vector w that is colinear with the given line. Any choice will do, say w = (1 , 2) T . The projection is v T w w T w w = 9 5 w = (9 / 5 , 18 / 5) T . 5.1.7 Find the distance from the point (1 , 2) to the line 4 x - 3 y = 0.
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sol51 - Partial Solution Set Leon Section 5.1 5.1.1c Find...

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