Test1_sol - NAME Spring 2011 MAC 2313 Test 1 UFID Section...

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Spring 2011, MAC 2313, Test 1 UFID: 2/3/2011, Section: 3124 1. Find the cosine of the angle between the normal vectors of the two planes as follows. (10 points) 2 x - y + 2 z = - 1 and 3 x + 4 y - 2 z = 5 . Solution. The normal vectors of the two planes are u = 2 i - j + 2 k and v = 3 i + 4 j - 2 k and hence the cosine of the angle θ between u and v is cos θ = u · v k u kk v k = 2 · 3 + ( - 1) · 4 + 2 · ( - 2) p 2 2 + ( - 1) 2 + 2 2 p 3 2 + 4 2 + ( - 2) 2 = - 2 9 29 = - 2 3 29 . 2. (1) Find a unit vector that is perpendicular to the plane which passes through points P (1 , 0 , 0), Q (2 , 0 , - 1) and R (1 , 4 , 3). (5 points) (2) Find the area of triangle PQR . (5 points) Solution. (1) First ~ PQ = i - j and ~ PR = 4 j + 3 k . Therefore a vector that is perpendicular to the plane passing through P , Q and R is v = ~ PQ × ~ PR = ± ± ± ± ± ± i j k 1 - 1 0 0 4 3 ± ± ± ± ± ± = 4 i - 3 j + 4 k and hence the normal vector is v k v k = 1 41 (4 i - 3 j + 4 k ). (2) The area of
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Test1_sol - NAME Spring 2011 MAC 2313 Test 1 UFID Section...

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