Exam 1 Solutions

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

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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 : 224 4 cos 0.314269 441111 6 31 8    uv , 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    uu v u vu u d) The vector projection of u in the direction of v Solution : 2 42 1,1, 4 18 9 v v e) The area of the parallelogram formed by u and v Solution : 9, 7,4 81 49 16 146 12.083     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      uvw 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 22 1 2 t xt yt zt      R .
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This note was uploaded on 02/10/2011 for the course MATH E-21a taught by Professor - during the Fall '09 term at Harvard.

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

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