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04win-m1

# 04win-m1 - MATH 51 MIDTERM 1 1 Find all solutions of the...

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MATH 51 MIDTERM 1 January 29, 2004 1. Find all solutions of the following system: x 1 - x 2 + x 3 + 2 x 4 = 3 x 2 + x 3 + x 4 = 3 x 1 + x 2 + 3 x 3 + 4 x 4 = 9 2. Let L be the intersection of the two planes x + y + z = 4 and 2 x + 3 y + z = 9 . Find a parametric equation for L . 3(a) Suppose u , v , and w are points in R n such that k u k = k v k = k w k = 1 and such that w = - u . Suppose also that v is not equal to u or to w . Prove that the triangle Δ uvw has a right angle at v . 3(b) Suppose x , y , and z are vectors in R n whose norms are 1, 2, and 3, respec- tively. Suppose each vector is orthogonal (i.e., perpendicular) to each of the other two. Find a scalar c such that the vector x + c y - z is orthogonal to the vector x + y + z . 4. Consider the points A = (1 , 1 , 1 , 1), B = (1 , 2 , 0 , - 1) and C = (1 , 0 , - 1 , 1) in R 4 . 4(a) Find the cosine of the angle at B of the triangle ABC . 4(b) Find a parametric equation for the plane through the points A , B , and C from part (a). 5. Are the following three vectors in R 3 linearly independent or linearly dependent?

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