M23Fall04a(sol)

M23Fall04a(sol) - Mathematics 23 Midterm Exam, September...

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Mathematics 23 Midterm Exam, September 30, 2004 + Solutions Answer all questions and be sure to show all work . No notes, books, or calculators are allowed. 1. Let a = i + j + 2k and let b = - 3i - 2j + 4k . Are a and b orthogonal, parallel, or neither? Explain carefully. Answer. Since a · b = - 1 , the vectors are not orthogonal. They are not parallel, since a is not a scalar multiple of b . So they are skew. 2. a. Find the unit tangent vector T ( t ) to the curve r ( t ) = h 6 t 5 , 4 t 3 , 2 t i . Answer. a. r 0 ( t ) = h 30 t 4 , 12 t 2 , 2 i and | r 0 ( t ) | = 900 t 8 + 144 t 4 + 4. So T ( t ) = ( 900 t 8 + 144 t 4 + 4) - 1 h 30 t 4 , 12 t 2 , 2 i . b. Find the parametric equations of the tangent line to r ( t ) at the point (6 , 4 , 2). Answer. t = 1 at the point (6 , 4 , 2). The direction numbers for the line are r 0 (1) = h 30 , 12 , 2 i . So the parametric equations become x ( t ) = 6+30 t,y ( t ) = 4+12 t, and z ( t ) = 2 + 2 t . 3. a. Find an equation of the plane
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M23Fall04a(sol) - Mathematics 23 Midterm Exam, September...

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