MATH 2050 Assignment 4 Fall 2014 Due: Monday, Oct. 20 [5] [10] −4 1 1. Let ~u = 2 and ~v = −3. Find the projection of ~u onto ~v ; and the projection of 3 4 ~v onto ~u respectively. 2. (a) Find two orthogonal vectors in the plane x + 3y − 2z = 0. 3 (b) Find the projection of ~u = −1 onto the plane x + 3y − 2z = 0. 1 [5] x −4 + t 3. Calculate the distance from point P (1, 1, 1) to the line y = 2 − t . z 3 + 2t [5] 4. Calculate the distance from point P (1, 1, 1) to the plane 5x − y + 2z = 1. [5] [10] 2 1 0 −1 5. Are vectors ~u = −1, ~v = 1, w ~ = 2, and ~x = 2 linearly independent? 1 2 1 1 1 1 0 6. Let ~u = 1 , ~v = 0 and w ~ = 1. 0 1 1 (a) Show that ~u , ~v and w ~ are linearly independent. (b) Show that any 3-dimensional vector can be written as a linear combination of ~u, ~v and w. ~ [10] −2 1 −1 7. Let ~u = 1 , ~v = −2, and w ~ = −1. 2 −1 1 (a) Are ~u , ~v and w ~ linearly independent. (b) Can any 3-dimensional vector be written as a linear combination of ~u, ~v and w? ~ If yes, please show your reason; if no, please provide one example. [50]
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