MATH 32A Study Guide - Midterm Guide: Parametric Equation, Cross Product
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Midterm 2 solution: find a vector normal to the plane with the given equation 5x 6y + z = 11. Ans: n = (cid:104)5, 6, 1(cid:105: find an equation of the plane passing through p = (1, 1, 1), q = (0, 1, 1), r = (4, 1, 2). P r is normal to the desire plane p. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 2i 5j 6k = (cid:104)2, 5, 6(cid:105). (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i j k. Now choose any one of the three points - say, q = (0, 1, 1) - and compute d. Oq = (cid:104)2, 5, 6(cid:105) (cid:104)0, 1, 1(cid:105) = 5 + 6 = 1. d = n . We conclude that p has equation 2x 5y 6z = 1: find the intersection of the line and the plane x 3y + 2z = 7, r(t) = (cid:104)0, 1, 1(cid:105) + t(cid:104)1, 0, 3(cid:105)
