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m1a-sol-20C-sp2006

# m1a-sol-20C-sp2006 - Name TA Name Math 20C Midterm Exam 1...

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Name: Section Number: TA Name: Section Time: Math 20C. Midterm Exam 1 April 28, 2006 No calculators or any other devices are allowed on this exam. Write your solutions clearly and legibly; no credit will be given for illegible solutions. Read each question carefully. If any question is not clear, ask for clarification. Answer each question completely, and show all of your work. 1. (a) (5 points) Find all constants c such that the vectors v = h 1 , c, 2 i and w = h c 2 , c, - 4 i are perpendicular to each other. (b) (5 points) Set c = 1 in vectors v and w above. In this case, find a unit vector perpendicular to both v and w . (c) (5 points) Keep c = 1. Find the scalar projection of v onto w . (a) 0 = v · w = h 1 , c, 2 i · h c 2 , c, - 4 i = c 2 + c 2 - 8 = 2 c 2 - 8 c 2 = 4 c = ± 2 . (b) c = 1 then v = h 1 , 1 , 2 i , w = h 1 , 1 , - 4 i , then u = i j k 1 1 2 1 1 - 4 = h ( - 4 - 2) , - ( - 4 - 2) , (1 - 1) i ⇒ u = h- 6 , 6 , 0 i , ⇒ | u | = 36 + 36 = 6 2 . Then a unit vector ˜ u normal to both v and w is ˜ u = u | u | = 1 2 h- 1 , 1 , 0 i .

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m1a-sol-20C-sp2006 - Name TA Name Math 20C Midterm Exam 1...

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