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Proof Lagrange's Identity

  1. Mar 2, 2013 #1
    Dear all,

    Any idea for the proof of the Lagrange's identity using tensor notations and Levi Civita symbol?

    (a x b).(c x d)=(a.c)(b.d) - (a.d)(b.c)

    x: cross product
    a,b,c,d: vectors

    Thanks
     
  2. jcsd
  3. Mar 2, 2013 #2
    Ok, thanks, in future i will be more careful.

    What about the dot product on the left side, how can i use Levi Civita symbol to represent it.
    Actually, the identity that you wrote and the cross product representation are all i know about the Levi Civita symbol but i couldn't use them.
     
  4. Mar 2, 2013 #3

    dx

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    Use the following identity:

    εijkεimn = δjmδkn - δjnδkm

    Also, in future, post questions like this in the homework section of PF, and tell us a little about how you've tried to solve the problem.
     
  5. Mar 2, 2013 #4

    dx

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    The i component of a x b is ajbkεijk and the i component of c x d is cmdnεimn

    So their dot product is ajbkcmdnεijkεimn
     
  6. Mar 2, 2013 #5
    ok from there,
    ajcjbkdk-ajdjbkck
    and i assume, similar form can be obtained for j and k components by just replacingg j s with k s, i s with j s and k s with i s. And in total i have 6 terms, 2 terms from each component. Am i right?

    But, on the right had side i think i should have more than 6 terms?
     
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