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Practice_Exam1_v_2

Practice_Exam1_v_2 - c3 18.02 Practice Exam 1A{50 mins...

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Unformatted text preview: c3 18.02)! Practice Exam 1A {50 mins.) ProbIEm l (25 pts: 5, 10, 10) The rectangular _box has edges of lengths 1,2,3 lying aiong the coordinate axes, as shown. Using head—to—tail addition, R0 = -2i + k. _ € 5.) Express similarly PO and PR in terms of i, j. and k . b) Find the cosine of the angle 6 = RPQ. c) Find a perpendicular vector to the plane through P, Q, and R, and find the area of triangle PQR. Problem 2. (25 pts: 10. 1o, 51 ,_ 1 0 '1 . . . . - 41/2 1 —1/2 a) If A = 2 l 2 ’ 31111 :ntthesmilslsing A-1 = _2 1 2 73 I" 1‘ IE - 3/2 b) Solve x1 + Ik3 = 2 if C = 3. using part (a). 2x1 + x2 +‘2x3 = 1 x1 + 2x2 + cx3 = 3 Then check the value'of _x by using Cramer’s ruler' {Show work.) 1 c) For one value of c the system in (b) has'no sclution. .Find this 'vaiue. Problem 3. (25 pts.: 5, 10, 10) _ A moVing point P has coordinates -x =. (t - 112. y =, t2. 2 = 2t - 1. ‘ a) Where (i.e., a_t what point) does it pass through the yz-plane? b) At time t = 0, find its speed and the unit tangent vector to the motion.- c) Find a constant vector perpendicular to the position vector R a 0?. What does this tell you geometrically about the motion? ' Problem 4. (10 pts.) Scotch tape is being peeled off a roll of radius a. starting at the point A. The end -P:(x,y) is always'puiied vertically upwards. Use vector methods to write parametric equations for x and y terms of 6, for Q s 9 s 1: (see fig. helm.) Problem 5. (10 pts.) Using. the standard operations on vectors. prove that if the diagonals of a parallelogram are perpendicular. its four sides have equal length. Problem 6. (5 pts). A point moves with constant speed. Prove its velocity vector v and itsacceleration vector a are always perpendicular. (Consider v-v .) P 0. H 9:2; ADG’ ...
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