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MAC2313 T1Solutions

MAC2313 T1Solutions - MAS 2313-03 Test 1 Fall 2007 Name...

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MAS 2313-03 Test 1 Fall 2007 Name: Show all work (No credit given if work is not shown). 7 points each unless indicated otherwise. 1. Write an equation of the sphere that passes through the point (1 , 1 , 2) and whose center is (2 , 3 , 4). The radius is the distance between (1 , 1 , 2) and (2 , 3 , 4), i.e. r = 1 + 4 + 4 = 3 Hence an equation for the sphere is ( x 2) 2 + ( y 3) 2 + ( z 4) 2 = 9. 2. Describe the set of all points ( x, y, z ) that satisfy x 2 + y 2 + z 2 6 x + 2 y 4 z = 0. Completing the square we get ( x 3) 2 + ( y + 1) 2 + ( z 2) 2 = 14 which describes the set of all points on a sphere of radius 14 and center C (3 , 1 , 2). 3. Write an equation for the set of all points ( x, y, z ) whose distance from (1 , 1 , 1) is the same as the distance from (2 , 0 , 3). (Simplify your equation). ( x 1) 2 + ( y 1) 2 + ( z 1) 2 = ( x 2) 2 + ( y 0) 2 + ( z 3) 2 or x 2 2 x + 1 + y 2 2 y + 1 + z 2 2 z + 1 = x 2 4 x + 4 + y 2 + z 2 6 z + 9 or x y + 2 z 5 = 0 (the equation of a plane). 4. Let −→ w = ( t, t, 2 ) , −→ v = ( 2 , t, 3 ) . Is there a value of t for which −→ w and −→ v are perpendicular ? Either find t or show carefully that there is no such t .
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