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Homework Help: Vector identity proof

  1. Mar 20, 2010 #1
    1. The problem statement, all variables and given/known data

    Hi. I need to prove that [b x c, c x a, a x b] = [a, b, c]2 for any three vectors a, b and c.

    Note that [a, b, c] = a(b x c)


    2. Relevant equations

    I tried using the identify (a x b) x c = (a.c)b - (a.b)c

    3. The attempt at a solution

    Using the above identity I got [b x c, c x a, a x b] = (b x c){(c x a) x (a x b)} = (b x c){(a(a x b)a - (ca)(a x b)}. I'm not sure where to go from here or if it's the correct approach at all

    Thanks for any help!
     
  2. jcsd
  3. Mar 20, 2010 #2

    CompuChip

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    Science Advisor
    Homework Helper

    If a, b, c are vectors, then (b x c) is a vector. So then, what does the multiplication a(b x c) mean? Is it inner product of a with (b x c)? Are the vectors three dimensional?
    Or are you working in a Clifford algebra where ab is the Clifford (geometrical) product and x is the commutator product (b x c = 1/2(bc - cb)).
     
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