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prelim1_spring2006

prelim1_spring2006 - Math 192 Prelim 1 Feb 21 2006...

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Math 192 Prelim 1, Feb. 21, 2006 Calculators are not allowed. Write your section number and TA’s name on the front of your workbook. This exam is worth 100 points. Point values for each problem are in parentheses. 1. Let u = - 2 i + 4 j - k and v = 2 i + 3 j - 6 k be two vectors in 3-space. a) (5) Find the angle between u and v. b) (4) Find a vector w of magnitude 5 units in the direction opposite to that of u + v. c) (4) Let u , v , and w be arbitrary vectors. Answer as True or False. (No reasons needed.) i) If u x v = u x w and u = 0, then v = w. ii) If u = 0 , u x v = u x w and u · v = u · w , then v = w. 2. a) (8) Find the line of intersection of the planes 2 x - y + 3 z = 4 and - x + y + 2 z = 1. Then find the plane perpendicular to this line and containing the point (5 , 6 , - 3) . b) (6) Find the distance between the two planes 2 x + 3 y - z = 4 and 2 x + 3 y - z = 9 . 3. Let r ( t ) = (cos t + t sin t ) i + (sin t - t cos t ) j be the position vector of a moving particle. a) (5) Find the length of the curve between t = π 2 and t = π.
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