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Math Help - Vector laws involving dot/cross product

  1. #1
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    Vector laws involving dot/cross product

    Please prove the following laws


    a.(bxc)= b.(cxa)=c.(bxa)
    ax(bxc)=b(a.c) -c(a.b)
    (axb).(cxd)= (a.c)(b.d) -(a.d)(b.c)
    ax(bxc) +bx(cxa) + cx(axb) = 0
    (axb)x(cxd) = b(a.(cxd))-a.(b.(cxd))
    = c(a.(bxd)) - d(a.(bxc)
    (axb).((bxc)x(cxa))= (a.(bxc))^2
    thank you!
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  2. #2
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    Re: Vector laws involving dot/cross product

    Try expanding out each vector in components, A = Ax(x) + Ay(y) + Az(z), and reorganizing to get your desired result. For example,

    A.(B X C) = A.[(ByCz - BzCy)(x) + (BzCx - BxCz)(y) + (BxCy - ByCx)(z)]
    = AxByCz - AxBzCy + AyBzCx - AyBxCz + AzBxCy - AzByCx
    = Bx(CyAz - CzAy) + By(CzAx - CxAz) + Bz(CxAy - CyAx)
    = B.(C X A)

    For the components of a cross product it's helpful to remember the following. Consider the cross product Q = R X S, which gives a vector Q. The x-component Q will involve cross terms of the non-x components of R and S, ie. RySz and RzSy. The positive term will be cyclic in (x,y,z) and the negative will be anticyclic. So, for example, in (RySz - RzSy)(x), the first term RySz(x) is cyclic (y,z,x), while the second term is anticyclic (z,y,x). You can verify this in the cross products above.
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  3. #3
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    Re: Vector laws involving dot/cross product

    Hello,
    Thanks for your reply! I tried expanding out each vector component and was told by my tutor that it was a waste of time and there were laws or rules that would make it much simpler.
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  4. #4
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    Re: Vector laws involving dot/cross product

    Why didn't your tutor do it for you?
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    Re: Vector laws involving dot/cross product

    She showed me how to one however I recently tried the other ones and i am still confused and was hoping seeing how the other ones were solved would help me understand how they were done. Any help would be appreciated
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  6. #6
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    Re: Vector laws involving dot/cross product

    Quote Originally Posted by Trianagt View Post
    She showed me how to one however I recently tried the other ones and i am still confused and was hoping seeing how the other ones were solved would help me understand how they were done. Any help would be appreciated
    This only a guess as to what your tutor may have meant.
    A\cdot(B\times C)=\left| {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\\{  {b_1}}&{{b_2}}&{{b_3}}\\{{c_1}} &{{c_2}}&{{c_3}}\end{array}} \right|
    Now by manipulating the rows of the determinate #1 is easy to see.
    Recall that A\times C=-(C\times A).
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  7. #7
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    Re: Vector laws involving dot/cross product

    Well I can't immediately think of how to prove the first two without expanding components. However, these first two identities can be used to prove all the rest.
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  8. #8
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    Re: Vector laws involving dot/cross product

    Quote Originally Posted by JohnDMalcolm View Post
    Well I can't immediately think of how to prove the first two without expanding components. However, these first two identities can be used to prove all the rest.
    If you look at reply #6, you will see that #1 is quite easily done without expansion.
    However, I agree that #2 must be done with expansion. It is a nightmare of an exercise in subscripts.
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