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Vector Algebra - Vector Triple Product Proof

  1. Nov 5, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove, by writing out in component form, that
    [tex]\left(a \times b \right) \times c \equiv \left(a \bullet c\right) b - \left(b \bullet c\right) a[/tex]


    and deduce the result, [tex]\left(a \times b\right) \times c \neq a \times \left(b \times c\right)[/tex], that the operation of forming the vector product is non-associative.




    3. The attempt at a solution
    So I took the LHS and made it into component form:

    [tex]LHS = \left(\left(\left(a_{z} \cdot b_{x} \cdot c_{z}\right) - \left(a_{x} \cdot b_{z} \cdot c_{z}\right)\right) - \left(\left(a_{x} \cdot b_{y} \cdot c_{y}\right) + \left(a_{y} \cdot b_{x} \cdot c_{y}\right)\right)\right) \hat{i} + \left(\left(\left(a_{x} \cdot b_{y} \cdot c_{x}\right) - \left(a_{y} \cdot b_{x} \cdot c_{x}\right)\right) - \left(\left(a_{y} \cdot b_{z} \cdot c_{z}\right) + \left(a_{z} \cdot b_{y} \cdot c_{z}\right)\right)\right) \hat{j}[/tex]
    [tex]+ \left(\left(\left(a_{y} \cdot b_{z} \cdot c_{y}\right) - \left(a_{z} \cdot b_{y} \cdot c_{y}\right)\right) - \left(\left(a_{z} \cdot b_{x} \cdot c_{x}\right) + \left(a_{x} \cdot b_{z} \cdot c_{x}\right)\right)\right) \hat{k} [/tex]

    [tex]RHS = \left(a_{x} \cdot b_{x} \cdot c_{x}\right) \hat{i} +\left(a_{y} \cdot b_{y} \cdot c_{y}\right) \hat{j} + \left(a_{z} \cdot b_{z} \cdot c_{z}\right) \hat{k} - \left(a_{x} \cdot b_{x} \cdot c_{x}\right) \hat{i} + \left(a_{y} \cdot b_{y} \cdot c_{y}\right) \hat{j} + \left(a_{z} \cdot b_{z} \cdot c_{z}\right) \hat{k}[/tex]

    This doesn't seem very difficult to me, however, the algebra of it is all over the place. I believe I found the component form of each side properly, but I could have screwed up somewhere. I am definitely not getting the RHS = LHS, so I am either screwing up somewhere or am totally on the wrong path. Any help, either noticing what I have done wrong, or a fresh start is much appreciated. Thanks.
     
    Last edited: Nov 5, 2009
  2. jcsd
  3. Nov 5, 2009 #2
    Recheck your RHS, all important terms are missing. Notice that a, b, and c have 3 components each, be careful in the multiplication. Also, you have problems on the signs in your LHS.
     
    Last edited: Nov 5, 2009
  4. Nov 5, 2009 #3

    lanedance

    User Avatar
    Homework Helper

    one thing i find useful for thrse problems is working in subcript notation, where repeated indicies mean a sum is performed over that index. The dot products & cross products then become:

    [tex] a \bullet b = a_i b_j \delta_{ij} = a_i b_i [/tex]
    [tex] (a \times b)_k = a_i b_j \epsilon_{ijk} [/tex]

    where [itex] \delta_{ij} [/itex] is the kronecker delat & [itex] \epsilon_{ijk} [/itex] is the levi cevita
    http://en.wikipedia.org/wiki/Kronecker_delta
    http://en.wikipedia.org/wiki/Levi-Civita_symbol

    takes a little time to learn this notation to start, but i find they're an extra tool in the toolbox & can save a lot of time
     
  5. Nov 5, 2009 #4
    I am not fully understanding the dot product multiplication rule because how I think it is supposed to be multiplied to show component form just keeps coming out as I have it shown above.

    Also, I fixed the signs of the LHS, I believe that is now correct.

    This is where I get stuck:

    [tex]RHS = \left(a_{x} \cdot c_{x} + a_{y} \cdot c_{y} + a_{z} \cdot c_{z}\right) b - \left(b_{x} \cdot c_{x} + b_{y} \cdot c_{y} + b_{z} \cdot c_{z}\right) a[/tex]
     
    Last edited: Nov 5, 2009
  6. Nov 5, 2009 #5
    In dot product multiplication you just have to multiply the components of the vectors with the same unit vectors (i.e. i to i, j to j, k to k).

    I can see in your solution above that you don't have a problem with the dot product multiplication. You just confused it (indicated by a dot) with the ordinary multiplication (indicated by the the parentheses).
     
  7. Nov 5, 2009 #6
    You are correct... I confused regular multiplication with dot product. But what exactly is the parentheses multiplication of vectors? I though that was the dot product. Still confused...
     
  8. Nov 5, 2009 #7
    Another thing, you have sign problems in your RHS as well, all the terms you have there should cancel out.
     
  9. Nov 5, 2009 #8
    Ya. Just don't understand from here...

    [tex]
    RHS = \left(a_{x} \cdot c_{x} + a_{y} \cdot c_{y} + a_{z} \cdot c_{z}\right) b - \left(b_{x} \cdot c_{x} + b_{y} \cdot c_{y} + b_{z} \cdot c_{z}\right) a
    [/tex]

    to the solution.
     
  10. Nov 5, 2009 #9
    Just do the normal multiplication like (a+x)(b+y) = ab + bx + ay + xy.
     
  11. Nov 5, 2009 #10
    Remember that b has three components; bx, by and bz. The same goes with a.
     
  12. Nov 5, 2009 #11
    If you are saying to multiply them like this...


    [tex]

    RHS = \left((a_{x} \cdot c_{x} \cdot b) + (a_{y} \cdot c_{y} \cdot b) + (a_{z} \cdot c_{z} \cdot b) \right) - \left((b_{x} \cdot c_{x} \cdot a) + (b_{y} \cdot c_{y} \cdot a) + (b_{z} \cdot c_{z} \cdot a)\right)

    [/tex]

    That really only makes me consider the same questions I was considering before -.-
     
  13. Nov 5, 2009 #12
    Exactly, now your b = bx i + by j + bz k and your a = ax i + ay j + az k. Substitute and simplify again. Remove the dots so you wont be confused it with the dot product.
     
  14. Nov 5, 2009 #13
    Actually an easy way to do this is suggested by lanedance, but it needs some more understanding of the tensor notations. Using the notations you're actually left working with indices.
     
  15. Nov 5, 2009 #14
    Luckily the easy way to do this also involves more learning. It's still not making sense btw.
     
  16. Nov 5, 2009 #15
    This one is almost close to the proof. Just continue this.
     
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