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fall01prelim2 - π √ 2 5(16 points Find the distance...

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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 parentheses. 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 . 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 ) 3. (16 points) Let f ( x, y, z ) = ye 2 x + y . Find f x , f y , and f z . 4. The position of a particle traveling in 3 dimensional space is given by the equation r ( t ) = (cos t ) i + (sin t ) j + t k . (a) (15 points) Find the distance travelled by the particle from time t = 0 to the time t = t 0 . (b) (5 points) Where is the particle when it has travelled a distance of
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Unformatted text preview: π √ 2? 5. (16 points) Find the distance between the two parallel planes. (The answer is not 9.) Plane1 : x + 2 y-3 z = 0 Plane2 : x + 2 y-3 z = 9 6. (16 points) The vector w is the mirror image of v re±ected over the dotted line as shown in the diagram below. The vector n is a unit vector perpendicular to the dotted line. Find a formula for w in terms of v and n . n v w B B Extra Credit: (10 points) Find the plane whose points are equidistant from the two parallel lines: L 1 : x = 1 + 2 t,y =-2 t,z =-3 + t L 2 : x = 4 + 2 s,y =-2 s,z = s...
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