MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable CalculusFall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.� � � � �18.02 Practice Exam 1 A Problem 1. (15 points) A unit cube lies in the first octant, with a vertex at the origin (see figure). −−� −−� a) Express the vectors OQ (a diagonal of the cube) and OR (joining O to the center of a face) in terms of ˆı, ˆ k.�, ˆ� �� ���� �� O Q �R z y x b) Find the cosine of the angle between OQ and OR. Problem 2. (10 points) The motion of a point P is given by the position vector R�= 3 cos t ˆ � + t ˆ ı + 3 sin t ˆ k. Compute the velocity and the speed of P . Problem 3. (15 points: 10, 5) ⎤ � ⎤ � a) Let A = ⎥ � 1 2 1 3 0 1 2 −1 0 � ⎣; then det(A) = 2 and A−1 = 1 2 ⎥ � 1 −1 2 a −2 2 b 5 −6 � ⎣; find a and b. ⎤ � ⎤ � x 1 ⎥ � ⎥�b) Solve the system A X = B, where X = � y ⎣ and B = � −2 ⎣ . z 1 c) In the matrix A, replace the entry 2 in the upper-right corner by c. Find a value of c for which the resulting matrix M is not invertible. For this value of c the system M X = 0 has other solutions than the obvious one X = 0: find such a solution by using vector operations. (Hint: call U, V and W the three rows of M, and observe that M X = 0 if and only if X is orthogonal to the vectors U, V and W .) Problem 4. (15 points) The top extremity of a ladder of length L rests against a vertical wall, while � the bottom is being pulled away. Find parametric equations for the midpoint P of the ladder, using as parameter the angle � between the ladder and the �P� horizontal ground. ���� Problem 5. (25 points: 10, 5, 10) a) Find the area of the space triangle with vertices P0 : (2, 1, 0), P1 : (1, 0, 1), P2 : (2, −1, 1). b) Find the equation of the plane containing the three points P0, P1, P2. c) Find the intersection of this plane with the line parallel to the vector V�= �1, 1, 1� and passing through the point S : (−1, 0, 0). Problem 6. (20 points: 5, 5, 10) a) Let R�= x(t)ˆ � + z(t)ˆı + y(t)ˆ k be the position vector of a path. Give a simple intrinsic formula for d (R�· R) in vector notation (not using coordinates).dtb) Show that if R�has constant length, then R�and V�are perpendicular. c) let A�be the acceleration: still assuming that R�has constant length, and using vector differ-entiation, express the quantity R�· A�in terms of the velocity vector
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