Pexam 1 - practice midterm 1 PDF

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MATH 53: Practice for 1st MIDTERM Problem 1. Let a and b be two vectors. a) What is the length of a × b in terms of |a|, |b|, and the angle between a and b. b) Show that (a × b) · (a × b) = |a|2 |b|2 − (a · b)2 . c) Find the equation of the plane through (0, 0, 0) containing lines described by vector equations r = h1, 2, 3it and r = th3, 2, 1i. d) What is is the distance of the point (1, 1, 1) to this plane? 2 2 2 Problem 2. Show that √ the surface x /8 + y + z /9 = 1, contains the curve x = 2 cos t , y = cos t/ 2 , z = 3 sin t . 3

Problem 3. Sketch the curve r = cos 2 θ , |θ | ≤ π/2 and find the area enclosed by it. Problem 4. Match the following curves described by vector valued functions to their plots: 0 ≤ t ≤ 10π and √ 1) hcos(t), sin(t), ti 2) hcos(t), sin(t), ti 3) hcos(t), sin(t), t2 i 4) hcos(t), sin(t), t3 i

Problem 5 a) Find the integral of the vector valued function in 1) of Problem 4 from 0 to 10π . b) Find the unit tangent vector to the curve in 3) of Problem 4 (for any value of t).

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