20e_04s_fs

20e_04s_fs - Math 20E Final Exam Solutions June 2004 1. Let...

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Unformatted text preview: Math 20E Final Exam Solutions June 2004 1. Let C = A B . By the definition of the cross product, C is perpendicular to A and C A is perpendicular to C and A . Since the length of a cross product of two perpendicular vectors is the product of their lengths, | C A | = | C || A | . 2. Since R /u = cos v i sin v j 2 u k and R /v = u sin v i u cos j , we have the normal N = R u R v = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle i j k cos v sin v 2 u u sin v u cos v vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = 2 u 2 cos v i + 2 u 2 sin v j u k . Since | N | = radicalbig 4 u 4 cos 2 v + 4 u 4 sin 2 v + u 2 = | u | 4 u 2 + 1 and u 0, the desired answer is N | N | = 2 u cos v i 4 u 2 + 1 2 u sin v j 4 u 2 + 1 + k 4 u 2 + 1 ....
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20e_04s_fs - Math 20E Final Exam Solutions June 2004 1. Let...

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