Given vectors u, v, and w, their scalar triple product is the value u . (v Times

Question

Given vectors u, v, and w, their scalar triple product is the value u . (v Times w). Show that if u = (u1, u2, u3),v = (v1,v2, v3), and w = (w1; w2,w3), then u . (v Times w) = u1v2w3 + u2v3w1 + u3v1w2 - u1v3w2 - u2v1w3 - u3v2w1 Prove that u . (v Times w) = (u Times v) . w. There are 27 possibilities for taking scalar triple product involving one or more of the basic vectors i, j, and k. In each case the result is either -1, 0, or 1. For each of those three values, indicate which of the 27 scalar triple products result in that value. Use the formula from 1 and show that A-{A Times s) = 0 and B . (A Times b) = 0 for vectors A and B. [Note that this shows that the cross product of two vectors is perpendicular to the original vectors if the original vectors are not parallel and each are non-zero.]

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Given vectors u, v, and w, their scalar triple product is the value u . (v Times

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