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Homework Help: (AxB).(CxD) = ?

  1. May 4, 2010 #1
    1. The problem statement, all variables and given/known data

    I am following along in a book and in one line the author asserts that

    [tex](\mathbf{A}\times\mathbf{B})\cdot(\mathbf{C}\times\mathbf{D}) = (\mathbf{A}\cdot\mathbf{C})(\mathbf{B}\cdot\mathbf{D}) - (\mathbf{A}\cdot\mathbf{D})(\mathbf{B}\cdot\mathbf{C})\qquad(1)[/tex]

    2. Relevant equations

    I believe that he is somehow using the rule that

    [tex]\mathbf{A}\times(\mathbf{B}\times\mathbf{C}) = \mathbf{B}(\mathbf{A}\cdot\mathbf{C}) - \mathbf{C}(\mathbf{A}\cdot\mathbf{B})\qquad(2)[/tex]

    3. The attempt at a solution

    Is this the only rule he is using to arrive at (1) ?
    I am having trouble see how to implement this to arrive at the same result. Am I missing something painfully obvious? :redface:
  2. jcsd
  3. May 4, 2010 #2


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    Homework Helper

    Did you try to do this in terms of components, i.e. using the definition of the cross and dot product? (Didn't try it myself, only suggesting.)

    Edit: although it might get a little messy...
  4. May 4, 2010 #3


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    Homework Helper

    I would think they'd use this formula as well as this one

    [tex]A\cdot (B \times C) = B \cdot (C \times A) = C \cdot (A \times B) [/tex]
  5. May 4, 2010 #4
    Ah yes, totally useful :smile:. Seeing as I have, in essence, a scalar triple product I would be hard pressed to start this without that rule :redface:

    I have solved it now.

    Thanks again! :smile:
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