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Math241_Fall08_2PM_Exam1 - Math 241 Exam 1 2PM V1 55 points...

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Math 241 – Exam 1 – 2PM V1 September 22, 2008 55 points possible 3. (a) (4pts) Find the angle between the vectors a = - 2 i + 2 j and b = 3 i + j + 4 k ). (b) (6pts) Find the area of the parallelogram formed by the vectors a = - 2 i + 2 j and b = 3 i + j + 4 k ). 6. (a) (3pts) State the dot product definition for the length of a vector in R n . (b) (3pts) Clearly state the Cauchy-Schwarz Inequality. (c) (4pts) Let a and b be vectors in R n . Use Cauchy-Schwarz to prove the Triangle Inequality || a + b || ≤ || a || + || b || . 4. Let A = 4 0 5 - 1 3 2 . (a) (3pts) Consider the function F ( ~x ) = A~x , F : R n R m . Clearly explain what the values of m and n are. (b) (3pts) Let ~a = (1 , 2 , 3). Compute F ( ~a ). (c) (4pts) Explain why the function F is a linear transformation. 2. (5pts) Find an equation of the plane containing the points ( - 1 , 0 , 4), (1 , - 5 , - 2) and (2 , - 4 , 8). 5. Let f ( x, y ) = - 2 x 2 - 3 y 2 . (a) (3pts) Draw three level sets of this function. Be sure to label the curves. (b) (2pts) Is it possible for two different level set to ever intersect. Briefly explain your answer.
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