exam 1 - 4 1 1 M = 1 3 2 2 2-1 3 1-1 A M-1 = 1 12 1 a b 1...

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18.02 Exam 1 Problem 1. Let P , Q and R be the points at 1 on the x -axis, 2 on the y -axis and 3 on the z -axis, respectively. ˆ a) (6) Express --! QP and QR --! in terms of ˆ i , j and k ˆ . b) (9) Find the cosine of the angle PQR . Problem 2. Let P = (1 , 1 , 1), Q = (0 , 3 , 1) and R = (0 , 1 , 4). a) (10) Find the area of the triangle PQR . b) (5) Find the plane through P , Q and R , expressed in the form ax + by + cz = d . c) (5) Is the line through (1 , 2 , 3) and (2 , 2 , 0) parallel to the plane in part (b)? Explain why or why not. Problem 3. A ladybug is climbing on a Volkswagen Bug (= VW). In its starting position, the the surface of the VW is represented by the unit semicircle x 2 + y 2 = 1, y 0 in the xy -plane. The road is represented as the x -axis. At time t = 0 the ladybug starts at the front bumper, (1 , 0), and walks counterclockwise around the VW at unit speed relative to the VW. At the same time the VW moves to the right at speed 10. a) (15) Find the parametric formula for the trajectory of the ladybug, and find its position when it reaches the rear bumper. (At t = 0, the rear bumper is at ( - 1 , 0).) b) (10) Compute the speed of the bug, and find where it is largest and smallest. Hint: It is easier to work with the square of the speed.
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Unformatted text preview: 4. 1 1 M = @ 1 3 2 2 2-1 3 1-1 A M-1 = 1 12 @ 1 a b 1 7-5 4-8 4 A (a) (5) Compute the determinant of M . b) (10) Find the numbers a and b in the formula for the matrix M-1 . x + 2 y + 3 z = 0 c) (10) Find the solution ~ r = h x, y, z i to 3 x + 2 y + z = t as a function of t . 2 x-y-z = 3 d ~ r d) (5) Compute . dt Problem 5. (a) (5) Let P ( t ) be a point with position vector ~ r ( t ). Express the property that P ( t ) lies on the plane 4 x-3 y-2 z = 6 in vector notation as an equation involving ~ r and the normal vector to the plane. d ~ r (b) (5) By di ↵ erentiating your answer to (a), show that is perpendicular to the normal vector dt to the plane....
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