sprelim1 - Mathematics 192 Fall 2006 Solutions to Prelim...

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Mathematics 192, Fall 2006: Solutions to Prelim Exam 1 1. (a) The vectors v = i + 2 j + a k and w = i + j + k are perpendicular if and only if v · w = 0 = 3 + a , so a = - 3. (b) The area of the parallelogram spanned by v and w is equal to | v × w | = | (2 - a ) i + ( a - 1) j - k | = 2 a 2 - 6 a + 6 This is equal to 6 if a = 0 or a = 3. 2. (a) Two vectors parallel to P are BA = 1 , 1 , 1 and CA = 0 , 0 , 2 . A normal vector is BA × CA = 2 , - 2 , 0 . A unit normal vector to the plane is therefore 1 2 1 , - 1 , 0 . Another correct answer is: - 1 2 1 , - 1 , 0 . (b) The equation of P is x - y = 1. A set of parametric equations for the line are x = 1 + t, y = 1 - t, z = 1. We substitute these into the equation for P and solve for t : (1 + t ) - (1 - t ) = 1, yielding t = 1 / 2. The point of intersection is (3 / 2 , 1 / 2 , 1). 3. (a) Compute v ( t ) = t j + t 2 k , so s ( t ) = t 0 | v ( u ) | du = t 0 u 1 + u 2 du = 1 3 (1 + t 2 ) 3 2 - 1 3 . The substitution v = 1 + u 2 is helpful in computing the integral. (b) The ratio | r ( t ) | s ( t ) = 1 + 1 4 t 4 + 1 9 t 6 1 3 (1 + t 2 ) 3 2 = 9 + 9 4 t 4 + t 6 (1 + t 2 ) 3 = t 6 + terms of lower degree t 6 + terms of lower degree tends to 1 as t → ∞ .
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