MATH 599 Final: MATH 5B UCSB 5B Fall 06 5bFinalSolutions
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1. (a) find parametric equations for the line that passes through the point (2, 0, 1) and is perpendicular to the plane with equation 4x y 2z = 1. The direction vector for this line is v = (4, 1, 2) and it must pass through the point (2, 0, 1). L1 : x = 1 + 2t y = 3 2t z = 3t. Since the plane contains the two lines, their direction vectors (1, 1, 0) and (2, 2, 3) are parallel to the plane. Hence their cross product will be a normal vector. = 3i 3j + 0k = ( 3, 3, 0). n = (1, 1, 0) (2, 2, 3) =(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j k i. To get a point in the plane, we can take any point in either line, so just set t = 0 in the equations for l1 to get the point (0, 2, 3).