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MATH 311 - Vector Analysis BONUS # 1: The parametric equation of the plane Π that goes through the point P~0 = (x0 , y0 , z0 ) and which ~1 = (a1 , b1 , c1 ), and is parallel to the plane defined by the non-colinear and non-zero vectors V ~ V2 = (a2 , b2 , c2 ) is given by the vector equation: ~ 1 + sV ~2 , P~ = P~0 + tV s, t ∈ IR. In components this vector equation becomes 3 equations: x = x0 + ta1 + sa2 y = y0 + tb1 + sb2 z = z0 + tc1 + sc2 Find a non-parametric equation of the plane not using cross-product, but instead by eliminating the parameters s, t in the above equations and reducing to just one equation that will involve the variables (x, y, z), and the data (x0 , y0 , z0 ), (a1 , b1 , c1 ), and (a2 , b2 , c2 ). If you are careful in your calculations you will obtain an equation of the form A(x − x0 ) + B(y − y0 ) + C(z − z0 ) = 0, where the coefficient A depends only on b1 , b2 , c1 , c2 , the coefficient B depends only on a1 , a2 , c1 , c2 , and the coefficient C depends only on a1 , a2 , b1 , b2 . ~1 × V ~2 . Show that the vector (A, B, C) is parallel to the cross-product V This Bonus is due on Wednesday August 31st, 2005. Please do it in a separate sheet from your usual homework since I will grade it, not the grader.