MATA23H3 Lecture Notes - Lecture 17: Cross Product, Parallelogram

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10 Mar 2016
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Cross product: let (cid:126)u = [u1, u2, u3] and (cid:126)v = [v1, v2, v3] be vectors in r3. The cross product of vectors (cid:126)u and (cid:126)v. = [u2v3 u3v2, u1v3 + u3v1, u1v2 u2v1: note (cid:126)u (cid:126)v is perpendicular to both (cid:126)u and (cid:126)v, example 2: let (cid:126)u = [1, 5, 2] and (cid:126)v = [ 3, 2, 4]. Calculate (cid:126)u (cid:126)v and verify that (cid:126)u (cid:126)v is perpendicular to both (cid:126)u and (cid:126)v. (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) (cid:20)(cid:12)(cid:12)(cid:12)(cid:12)5 2 (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) 1 (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) 1. = 16 50 + 34 = 0. ((cid:126)u (cid:126)v) (cid:126)v = [16, 10, 17] [ 3, 2, 4] = 48 20 + 68 = 0: in r3, the area of a parallelogram determined by (cid:126)u = [u1, u2, u3] and (cid:126)v = [v1, v2, v3] is: Area = ||(cid:126)u (cid:126)v|| = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20)(cid:12)(cid:12)(cid:12)(cid:12)u2 u3 v3 v2 (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12)u1 u3 v1 v3 (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12)u1 u2 v1 v2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

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