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Math 2263 Multivariable Calculus Quiz 1: 12.4-12.5 Name: June 16, 2011 1. (a) Find two unit vectors orthogonal to both h0, 1, 2i and h1, −1, −2i. Solution: The cross product of the vectors is orthogonal to both of them. h0, 1, 2i × h1, −1, 2i = h0, 2, −1i √ √ 2 5 5 The unit vector in this direction is 0, 5 , − 5 . To get two unit vectors, we’ll use this vector and the one in the opposite direction, √ √ √ √ 2 5 5 2 5 5 0, ,− , and 0, − . 5 5 5 5 (b) Use part of your work from (a) to find an equation for the line of intersection of the planes y + 2z = 1 and x = 1 + y + 2z. Solution: We find a point in the line by inspection. One possible point is r0 = h2, 1, 0i. We find a direction vector by finding a vector orthogonal to both normal vectors. v = h0, 1, 2i × h1, −1, 2i = h0, 2, −1i The line equation is r = h2, 1, 0i + th0, 2, −1i. 2. Are the following lines parallel, skew, or intersecting? Line 1: x = 4t, y = 2 − 4t, z = 6t + 5 Line 2: x = −6s, y = 6s + 3, z = 2 − 9s Solution: Their directions vectors are scalar multiples, so they are parallel.