midterm03 - Math 2263 Spring 2008 Midterm 1 WITH SOLUTIONS...

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Math 2263 Name (Print): Spring 2008 Student ID: Midterm 1, WITH SOLUTIONS Section Number: February 21, 2008 Teaching Assistant: Time Limit: 50 minutes Signature: 1 15 pts 2 15 pts 3 15 pts 4 15 pts 5 10 pts 6 15 pts 7 15 pts TOTAL 100 pts 1. (15 points) Find an equation for the plane passing through all three points x, y, z = 3 , - 2 , - 2 , 2 , 0 , 1 and 1 , 0 , 0 . SOLUTION: A normal vector v is the cross product of 3 , - 2 , - 2 - 1 , 0 , 0 and 2 , 0 , 1 - 1 , 0 , 0 . So v = i j k 2 - 2 - 2 1 0 1 = - 2 i - 4 j + 2 k. We can divide v by - 2, so an equation for the plane is ( x - 1) + 2( y - 0) - ( z - 0) = 0, equivalently x + 2 y - z = 1 . 2. (15 points) Find an equation for the surface in ( x, y, z )-space obtained by rotating the hyper- bola x 2 - 4 z 2 = 1 of the ( x, z )-plane about the x -axis. SOLUTION: | z | is the distance form the x -axis in the ( x, z )-plane; we want to replace it with the distance to the x -axis in space, namely y 2 + z 2 . The equation of the surface of revolution is x 2 - 4 y 2 - 4 z 2 = 1 .
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Math 2263 Spring 2008 Midterm 1, WITH SOLUTIONS- Page 2 of 3 February 21, 2008 3. (15 points) The lines given paramerically by ( x, y, z ) = (7 + 2 t, - 1 - t, - 2 t ) , -∞ < t < and ( x, y, z ) = (4 -
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