Exam_solutions_1_(2)

Exam_solutions_1(2) - Mathematics 241 First Exam Solutions Dr Rosenberg Friday 1 a Show that the points P1 =-1-2 6 P2 =(0 4 1 P3 =(1 0 1 determine

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Mathematics 241 First Exam Solutions Dr. Rosenberg Friday, February 28, 2003 1. a) Show that the points P 1 = ( - 1 , - 2 , 6) , P 2 = (0 , 4 , 1) , P 3 = (1 , 0 , 1) determine a unique plane P , and determine the equation of P . Solution: To show they determine a unique plane, it’s enough to show that the line segments P 1 P 2 and P 1 P 3 are not parallel. But ( P 2 - P 1 ) × ( P 3 - P 1 ) = P 2 × P 3 - P 1 × P 3 + P 2 × P 1 = (4 , 1 , - 4) + ( - 26 , 2 , - 4) - (2 , 7 , 2) = ( - 20 , - 5 , - 10) = - 5(4 , 1 , 2) 6 = (0 , 0 , 0) . So the points are not collinear and (4 , 1 , 2) is perpendicular to P . Then the equation of P is (4 , 1 , 2) · ( x, y, z ) = (4 , 1 , 2) · ( - 1 , - 2 , 6) or 4 x + y + 2 z = 6 . b) Find the area of the triangle with vertices P 1 , P 2 , and P 3 . Solution: 1 2 ± ± ( P 2 - P 1 ) × ( P 3 - P 1 ) ± ± = 5 2 ± ± (4 , 1 , 2) ± ± = 5 2 21 . 2. (10 points) Let P 1 and P 2 be the planes with equations 3 x - 4 y + z = 2 , 3 x + 2 y - z = 7 . Find symmetric equations of the line
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This note was uploaded on 09/09/2009 for the course MATH 241 taught by Professor Wolfe during the Spring '08 term at Maryland.

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Exam_solutions_1(2) - Mathematics 241 First Exam Solutions Dr Rosenberg Friday 1 a Show that the points P1 =-1-2 6 P2 =(0 4 1 P3 =(1 0 1 determine

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