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Math Help - simplify ax[ax(axb)]dotc

  1. #1
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    simplify ax[ax(axb)]dotc

    Hello, we were given a few practice problems in class the other day and one of them asks...

    Simplify a x [a x (a x b)] dot c (where the x refers to cross product)

    I have tried to simplify it but have ended up with about a page and a half of computation that I don't know what to do with.

    I imagine there is a trick to this but I am clueless. If anyone can help I would appreciate it.
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  2. #2
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    Re: simplify ax[ax(axb)]dotc

    Hey lytwynk.

    Hint: Use the fact that the cross product is associative and re-group [a x (a x b)] = (a x a) x b.
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  3. #3
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    Re: simplify ax[ax(axb)]dotc

    But I thought the cross product was not associative for example a x (b x c) does not equal (a x b) x c... At least that's what my text says.
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  4. #4
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    Re: simplify ax[ax(axb)]dotc

    Sorry you are right: it is not associative.

    Try using the Jacobi identity and you should get some cancellations:

    Jacobi identity - Wikipedia, the free encyclopedia
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  5. #5
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    Re: simplify ax[ax(axb)]dotc

    You could also use the identity A \times (B \times C) = (A \cdot C)B - (A \cdot B)C.

    - Hollywood
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