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Mechanics and Arbitrary Vectors

Show that:
a. Sum of vectors i and j: epsilon(ijk)delta(ij) =0
b. Sum of vectors j,k: epsilon(ijk)epsilon(njk) = 2delta (in)
c Sum of vectors i,j,k: epsilon(ijk)epsilon(ijk) = 6

Show that:
Sum of vectors k: epsilon (jk)epsilon (mnk) = delta(im)delta (jn) - delta (in)delta (jm)

Using the epsilon (ijk) notation and in full explicit detail derive:
(AxB) . (CxD) = (A . C) (B . D) - (A.D)(B.C)

Let A be an arbitrary vector and let e be a unit vector in some fixed direction. Show that:
A = e (A.e) + ex (A x e)


Solution Summary

This handwritten solution contains step-by-step vector calculations to prove the vector sum statements in the question set. Brief explanations are included, and all formulas are clearly shown.