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calc3_vectorsstest - 18.02 Spring 2008 Practice Test 1 The...

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18.02, Spring 2008 Practice Test 1 The first test will take place on Thursday, 21 February and will cover the material of lectures one to six. The problems below should give you a sense of the level of difficulty and length of the test—you do not need to turn them in. The solutions will be made available to you on Monday, 18 February. There will be a review session for the test on Wednesday, 20 February at 7:30 p.m. in 4-163. 1. (a) Find the area of the triangle with vertices P 1 = (2 , 1 , 0), P 2 = (1 , 1 , 1), and P 3 = (0 , - 1 , 3). (b) Find the equation of the plane containing the points P 1 , P 2 , and P 3 . (c) Find the point where the line through Q 1 = ( - 2 , 1 , 3) and Q 2 = (0 , - 3 , 1) intersects the plane found in (b). 2. Find all the values of c for which the planes 2 x - y + z = 0 4 x - 2 y + z = 0 cx + y = 1 meet in a unique point. 3. (a) Find the point of intersection of the three planes 2 x + y - 2 z = 1 x - y - z = 3 x + 2 y + z = 2 by writing the equations above as a linear system Ax = b , finding the inverse of A , and computing A - 1 b .
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