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m1sol-20C-su2004

m1sol-20C-su2004 - Name Math 20C Midterm Exam 1 July 9 2004...

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Name: Student Number: Math 20C. Midterm Exam 1 July 9, 2004 Read each question carefully, and answer each question completely. Show all of your work. No credit will be given for unsupported answers. Write your solutions clearly and legibly. No credit will be given for illegible solutions. 1. (4 points) Consider the vectors ~v = 2 ~ i - 2 ~ j + ~ k and ~w = ~ i + 2 ~ j - ~ k . (a) Compute ~v · ~w . ~v · ~w = h 2 , - 2 , 1 i · h 1 , 2 , - 1 i = 2 - 4 - 1 = - 3 . (b) What is the cosine of the angle between ~v and ~w ? | ~v | = 4 + 4 + 1 = 3 , | ~w | = 1 + 4 + 1 = 6 . cos( θ ) = ~v · ~w | ~v | | ~w | = - 3 3 6 = - 1 6 . (c) Find a unit vector in the direction of ~v . ~u = ~v | ~v | = 1 3 h 2 , - 2 , 1 i . # Score 1 2 3 4 Σ
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2. (4 points) Find the equation of the plane that contains the lines ~ r 1 ( t ) = h 1 , 2 , 3 i t and ~ r 2 ( t ) = h 1 , 1 , 0 i + h 1 , 2 , 3 i t . P 0 = (1 , 1 , 0) is in the plane. P 1 = (1 , 2 , 3) = ~ r 1 ( t = 1) is also in the plane. Therefore, ~ P 0 P 1 = h 0 , 1 , 3 i is tangent to the plane. ~v = h 1 , 2 , 3 i is also tangent to the plane. Then, the normal vector to the plane ~n can be computed as follows: ~n = ~v × ~ P 0 P 1 = ~ i ~ j ~ k 1 2 3 0 1 3 = (6 - 3) ~ i - (3 - 0) ~ j + (1 - 0) ~ k = h 3 , - 3 , 1 i .
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