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# math1 - FIRST PRACTICE MIDTERM The following ve questions...

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FIRST PRACTICE MIDTERM The following five questions are worth 20 points each, although many have multiple parts. Show ALL your work in your answers to each question. You have 75 minutes for the examination. Calculators are NOT permitted. 1. Let P, Q, and R be points in R 3 with coordinates P = (0 , 2 , 1), Q = (2 , 0 , 1), R = (0 , 0 , - 1). 1a. Give a parametrization (in terms of auxiliary variables t and u ) for the plane containing P , Q , and R . 1b. Give an equation of the form ax + by + cz = d for the plane containing P , Q , and R . 2. Draw the level sets f ( x, y ) = c for the given values of c , in the given domains. 2a. f ( x, y ) = | x | + | y | , - 2 x 2, - 2 y 2, c = - 1 , 0 , 1 , 2 , 3 , 4. 2b. f ( x, y ) = xy , - 2 x 2, - 2 y 2, c = - 2 , - 1 , 0 , 1 , 2. 3. Define vectors ~u , ~v in R 3 by ~u = ( - 2 , 1 , 0), ~v = (0 , 2 , - 1). Let θ be the angle between the two vectors. 3a. Compute the lengths of ~u and ~v . 3b. Compute cos θ . 3c. Give a unit vector orthogonal to both
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