exam 2 - 18.02 Practice Exam 1 z Problem 1(15 points A unit...

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? ~ 18.02 Practice Exam 1 Problem 1. (15 points) A unit cube lies in the first octant, with a vertex at the origin (see figure). a) Express the vectors --! OR OQ (a diagonal of the cube) and --! (joining O to the center of a face) in terms of ˆ ı, ˆ k. , ˆ O Q 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) 3 2 3 2 1 3 2 1 a b A - 1 = 1 a) Let A = 6 4 7 5 6 4 7 5 ; find a and b . 2 0 - 1 0 ; then det( A ) = 2 and - 1 - 2 5 2 2 2 1 1 - 6 3 2 3 2 x 1 6 4 y 7 5 and B = 6 4 7 5 b) Solve the system A X = B , where X - 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
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