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Math241_Spring08_Exam1

# Math241_Spring08_Exam1 - Math 241 Exam 1 Version 1 50...

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Math 241 – Exam 1 – Version 1 September 26, 2007 50 points possible 1. Consider the points A (5 , - 4 , 7) and B (3 , 0 , 1). (a) (3pts) Find a vector equation or a set of parametric equations that defines the line containing points A and B . (b) (2pts) Determine if the line from part (a) intersects the plane 3 x + 2 y - 7 z = - 86. 2. (a) (2pts)If a = (1 , - 1 , 7 , 3 , 2) and b = (2 , 5 , 0 , 9 , - 1), calculate proj b a . (b) (3pts) Under what circumstances is proj a b = proj b a ? Briefly explain your answer. 3. (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 || . (Hint: Start with || a + b || 2 .) 4. Let u = (1 , 2 , - 1), v = (3 , 0 , - 1) and w = ( - 5 , 4 , b ). (a) (3pts) Find the volume of the parallelopiped determined by u , v and w . (b) (2pts) For what values of b are the vectors u , v and w coplanar? 5. Consider the lines l 1 : x = t - 3 , y = 1 - 2 t, z = 2 t + 5 and l 2 : x = 4 - 2 t, y = 4 t + 3 , z = 6 - 4 t .

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