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practice midterm 1

# practice midterm 1 - SOLUTIONS TO 18.02 PRACTICE MIDTERM#1...

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SOLUTIONS TO 18.02 PRACTICE MIDTERM #1 BJORN POONEN October 1, 2009 1) Let A , B , and P be points in space such that P is on the line segment AB and the distance AP is twice the distance BP . Find the position vector P in terms of the position vectors A and B . Solution: Let r ( t ) be the position of a particle moving with constant speed along ←→ AB so that it is at A at t = 0 and at B at t = 1. Then P is the point at time t = 2 / 3 (since 2 / 3 = 2(1 - 2 / 3)). We have r ( t ) = A + t ( B - A ), so r (2 / 3) = 1 3 A + 2 3 B . 2) What is the upper right entry of 6 5 4 3 2 1 0 1 - 2 3 0 - 4 5 - 6 0 T ? Solution: It is the dot product of the first row of the first matrix with the last column of the transposed second matrix. That column is the the last row of the untransposed second matrix. So the answer is 6 , 5 , 4 · 5 , - 6 , 0 = 0 . 3) Let a be a nonzero vector. What is the geometric shape formed by the set of points P in space whose position vector P satisfies | P × a | = 1?

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