Exam 1 Solutions - Math E-21a Exam#1 solutions Fall 2009 1...

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1 Math E-21a Exam #1 solutions – Fall 2009 1) Given the vectors u = 2, 2,1 , v = 1,1,4 , and w = 2,1,2 , find the following quantities and complete the box . Show all work below. a) The angle between the vectors u and v Solution : 2 2 4 4 cos 0.314269 4 4 1 1 1 16 3 18 u v u v , so 1 cos (.314269) 71.68 b) The scalar projection of v in the direction of u Solution : 4 3 u v u c) The vector projection of v in the direction of u Solution : 2 4 2, 2,1 9   u u v u v u u u u d) The vector projection of u in the direction of v Solution : 2 4 2 1,1,4 1,1,4 18 9 u v v v e) The area of the parallelogram formed by u and v Solution : 9, 7,4 81 49 16 146 12.083 u v f) The volume of the parallelepiped with u , v , and w as three edges Solution : 9, 7,4 2,1,2 18 7 8 17 17     u v w 2) Let M = (2,1,2) , N = (4,2,1) , P = (3,1, 1) , and Q = (2,4,5) be points in R 3 . Find the following: a) Find parametric equations for the line passing through the points M and N. Solution : First, calculate the direction vector MN 2,1, 1 v  . Using the initial point M = (2,1,2) , we get the parametric equations 2 2 1 2 t x t y t z t
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