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Math206Test1

# Math206Test1 - Mid-term Exam 1 MthSc 206 013 Calculus of...

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Mid-term Exam # 1 September 24, 2007 MthSc 206 – 013: Calculus of Several Variables Instruction: PRINT your name clearly in capitals and underline your last name . NAME: EXAM SOLUTION 1. (3pts). Show that the following equation represents a sphere and find its center and radius: x 2 + y 2 + z 2 + 2 x - 6 y + 4 z + 10 = 0 . Solution: x 2 + 2 x + y 2 - 6 y + z 2 + 4 z = - 10 x 2 + 2 x + 1 + y 2 - 6 y + 9 + z 2 + 4 z + 4 = - 10 + 1 + 9 + 4 ( x + 1) 2 + ( y - 3) 2 + ( z + 2) 2 = 4 It is an equation for a sphere. The center is c ( - 1 , 3 , - 2) and its radius is 2 . 2. (8pts) Two vectors a = 1 , 2 , 3 and b = 0 , 1 , 0 are given. (a) (3pts) Find the scalar projection (scalar component) of b onto a . Solution: Comp a b = a b | a | = 1 , 2 , 3 0 , 1 , 0 q (1) 2 + ( 2) 2 + ( 3) 2 = 2 6 = 3 3 = 0 . 577 . (b) (3pts) Find the vector projection of b onto a . Solution: Proj a b = Comp a b · a | a | = 3 3 · 1 , 2 , 3 6 = 1 3 2 1 , 2 , 3 (c) (2pts) Find the angle between the vectors a and b . Solution: Since a b = | a | | b | cos θ , we have 1 , 2 , 3 0 , 1 , 0 = 6 · 1 · cos θ 2 = 6 · 1 · cos θ.

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