Winter 2006 - Hall's Class - Practice Exam 1

Winter 2006 - Hall's Class - Practice Exam 1 - Math 20C...

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Math 20C Practice Midterm I Solutions January 27, 2006 1. (a) (10 points) There are two unit vectors pointing in the direction of the line y = 3 x . Find the one with positive x -component. The points (0,0) and (1,3) are on the line, so the vector a = h 1 - 0 , 3 - 0 i = h 1 , 3 i is a vector in the direction of the line. | a | = 1 + 3 2 = 10, so a | a | = h 1 10 , 3 10 i is a unit vector in the direction of the line. Since the x component is positive, this is the vector we are looking for. (b) (10 points) Let A = (2 , 2) and B = (3 , 0). Find the scalar projection of -→ AB onto the vector from part (a). Illustrate your answer with a diagram. The vector -→ AB is h 3 - 2 , 0 - 2 i = h 1 , - 2 i . The scalar projection of -→ AB onto a is ( -→ AB · a ) | a | = (1 - 6) 10 = - 5 10 . 2. (a) (5 points) Find the center and radius of the sphere x 2 + 2 x + y 2 - 4 y + z 2 = 8 . We need to rewrite this equation in the form ( x - x 0 ) 2 + ( y - y 0 ) 2 + ( z - z 0 ) 2 = r 2 , a sphere of radius r and center ( x 0 ,y 0 ,z 0 ). We do this by completing the square of x 2 + 2 x and y 2 - 4 y : x 2 + 2 x = x 2 + 2 x + 1 - 1 = ( x + 1) 2 - 1 y 2 -
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This note was uploaded on 04/29/2008 for the course MATH 20C taught by Professor Helton during the Winter '08 term at UCSD.

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Winter 2006 - Hall's Class - Practice Exam 1 - Math 20C...

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