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01prelim2

# 01prelim2 - Prelim 2 November 1 2001 Show all your work...

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Prelim 2 November 1, 2001 Show all your work. Your reasoning is important. No calculators are allowed. You are allowed to use a 5 by 8 index card. The number of points each problem is worth is indicated in parantheses. Good luck. 1. (16 points) Express the vector 6 i + 2 j as the sum of 2 vectors a and b such that a is parallel to the vector i + j and b is perpendicular to i + j . SOLUTION: Let’s name our vectors; v = i + j and u = 6 i + 2 j . Now project u onto v to get: proj v ( u ) = u · v | v | 2 · v = 6 + 2 2 ( i + j ) = 4 i + 4 j So now we have u = (4 i + 4 j ) + (2 i 2 j ) with the vector 4 i + 4 j parallel to v and the vector 2 i 2 j perpendicular to v . 2. (16 points) Find the plane that contains the point P (1 , 2 , 1) and the line: r ( t ) = t (2 i + 4 j + k ) + (2 i + j + 3 k ) SOLUTION: The normal vector to the plane we want is: n = i j k 2 4 1 1 3 2 = i 4 1 3 2 j 2 1 1 2 + k 2 4 1 3 = (8 3) i (4 1) j + (6 4) k = 5 i 3 j + 2 k So the equation of our plane is: 5( x 1) 3( y + 2) + 2( z 1) = 0 3. (16 points) Let f ( x, y, z ) = ye 2 x + y . Find f x , f y , and f z .

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01prelim2 - Prelim 2 November 1 2001 Show all your work...

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