week01 - Math 23 Spring 2005 B Dodson J Mohler 1 Course Info 2 Week 1 Homework 12.1 12.2 Distance vectors Problem 12.1.15 Show that the equation x2 y 2

# week01 - Math 23 Spring 2005 B Dodson J Mohler 1 Course...

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Math 23, Spring 2005 B. Dodson J. Mohler 1. Course Info 2. Week 1 Homework: 12.1, 12.2 Distance, vectors Problem 12.1.15: Show that the equation x 2 + y 2 + z 2 - 6 x + 4 y - 2 z = 11 represents a sphere, and find its center and radius. Solution: ( x - a ) 2 = x 2 - 2 ax + a 2 , so the terms x 2 - 6 x = ( x - 3) 2 - 9 . Likewise, y 2 + 4 y = . . . z 2 - 2 z = . . . y 2 + 4 y = ( y + 2) 2 - 4 z 2 - 2 z = ( z - 1) 2 - 1, so
2 x 2 + y 2 + z 2 - 6 x + 4 y - 2 z = ( x - 3) 2 - 9 + ( y + 2) 2 - 4 + ( z - 1) 2 - 1 = 11 , or ( x - 3) 2 + ( y + 2) 2 + ( z - 1) 2 = 11 + 9 + 4 + 1 = 5 2 . conclusion: center = (3, -2, 1), radius = 5. Problem 12.2.19a: Find | a | and a - 2 b when a = < 6 , 2 , 3 >, b = < - 1 , 5 , - 2 > . Solution: The length | a | = 36 + 4 + 9 = 7 , and a - 2 b = < 6 , 2 , 3 > - 2 < - 1 , 5 , - 2 > = < 8 , - 8 , 7 > .