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pract_mt_1 - Weds Jan 30 2008 Page 1/9 MATH 32A FIRST...

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Weds Jan. 30, 2008 Page 1/9 MATH 32A: FIRST MIDTERM EXAMINATION Winter 2008 1
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Weds Jan. 30, 2008 Page 2/9 1. (20 points) Find a vector valued function that represents the curve of inter- section of the cylinder x 2 + y 2 = 16 and the plane x + z = 5. Graph the curve and label the orientation corresponding to your parametrization. 2
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Weds Jan. 30, 2008 Page 3/9 2. (20 points) Let P1: x + y - z = 1 and P2: 2 x - 3 y + 4 z = 5 be 2 planes in R 3 . (a) Show that the planes are neither parallel nor perpendicular. (b) Find the equation of the line where the planes intersect. (c) Find the angle between the two planes. (You may use the back of this sheet if additional space is needed.) 3
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Weds Jan. 30, 2008 Page 4/9 3. (20 points) Let C be the curve defined by the parametric equations r ( t ) = < t 2 , 2 t 3 / 2 , 3 t 4 / 3 > . (a) Use vector calculus to verify that x = 1 + 2 t, y = 2 + 3 t, z = 3 + 4 t are the parametric equations to the tangent line L of C at the point (1 , 2 , 3) (e.g. when t = 1). (b) Does the tangent line L from (a) intersect the line L 2 defined by the parametric equations x = - 1 + 6 s, y = 3 - s, z = - 5 + 2 s ? Show your work.
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