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fsol-20C-su2004

fsol-20C-su2004 - Name(Use capitals Student number Math 20C...

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Name: (Use capitals) Student number: Math 20C Final Exam July 31, 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. (8 points) (a) Find the angle between the planes 2 x + y + 3 z = 1 and - x - 3 y + 2 z = 5. ~n 1 = h 2 , 1 , 3 i , ~n 2 = h- 1 , - 3 , 2 i , | ~n 1 | = 4 + 1 + 9 = 14 , | ~n 2 | = 4 + 1 + 9 = 14 . cos( θ ) = ~n 1 · ~n 2 | ~n 1 | | ~n 2 | = - 2 - 3 + 6 14 = 1 14 . So the answer is cos( θ ) = 1 14 . (b) Find a vector parallel to the line of intersection of the planes given in (a). ~v = ~n 1 × ~n 2 = ~ i ~ j ~ k 2 1 3 - 1 - 3 2 = h (2 + 9) , - (4 + 3) , ( - 6 + 1) i , ~v = h 11 , - 7 , - 5 i . # Score 1 2 3 4 5 6 7 8 Σ 1
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2. (8 points) A particle has the velocity function ~v ( t ) = h- 3 sin( t ) , 4 , 3 cos( t ) i . (a) Find the particle acceleration ~a ( t ). ~a ( t ) = ~v ( t ) 0 = h- 3 cos( t ) , 0 , - 3 sin( t ) i . (b) The particle initial position is given by ~ r (0) = h 3 , 1 , 0 i . Find the particle position function ~ r ( t ). ~ r ( t ) = h 3 cos( t ) + x 0 , 4 t + y 0 , 3 sin( t ) + z 0 i , ~ r (0) = h 3 + x 0 , y 0 , z 0 i = h 3 , 1 , 0 i , x 0 = 0 , y 0 = 1 , z 0 = 0 .
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