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Unformatted text preview: Copyright: Dr. Frederick Park Weds Jan. 30, 2008 Page 1/6 MATH 32A: FIRST MIDTERM EXAMINATION Winter 2008 SOLUTIONS Disclaimer: Many Problems will have multiple solutions. 1 Copyright: Dr. Frederick Park Weds Jan. 30, 2008 Page 2/6 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. Solution: First off, clearly the intersection will look like an ellipse. Now, we note that the projection of the curve C onto the xy plane is a circle or radius 4 given by x 2 + y 2 = 4 2 . A parametrization for this curve would be x = 4cos t , y = 4sin t . Now, x and y must also satisfy x + z = 5. Thus, z = 5 x = 5 4cos t . Therefore, a vector valued function would be r ( t ) = < 4cos t, 4sin t, 5 4cos t > and the curve C will look like an ellipse oriented in the counterclockwise direction. 2 Copyright: Dr. Frederick Park Weds Jan. 30, 2008 Page 3/6 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.) Solution: This problem is exactly as on the practice midterm. (a) n 1 = < 1 , 1 , 1 > is the normal vector to P1 and n 2 = < 2 , 3 , 4 > is the normal vector to P2. Then n 1 n 2 = < 1 , 6 , 5 > . Thus,  n 1 n 2  =  < 1 , 6 , 5 >  6 = 0 implying that P1 and P2 are not parallel....
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This note was uploaded on 05/15/2008 for the course MATH 32A taught by Professor Gangliu during the Spring '08 term at UCLA.
 Spring '08
 GANGliu
 Math

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