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Homework Help: Help? Linear algebra proof

  1. Apr 7, 2010 #1
    Help!? Linear algebra proof

    1. The problem statement, all variables and given/known data
    Suppose that u,v,w are geometric vectors such that u[tex]\neq[/tex]0,
    u[tex]\cdot[/tex]v=u[tex]\cdot[/tex]w and uxv=uxw

    Prove that v=w


    2. Relevant equations



    3. The attempt at a solution
    So far, I'm not sure if this is correct
    u[tex]\cdot[/tex]v=u[tex]\cdot[/tex]w
    |u||v|cos[tex]\theta[/tex]=|u||w|cos[tex]\theta[/tex]
    |v|=|u|


    uxv=uxw
    |u||v|sin[tex]\theta[/tex][tex]\hat{(u\times v)}[/tex]=|u||w|sin[tex]\theta[/tex][tex]\hat{(u\times w)}[/tex]
    |w|sin[tex]\theta[/tex][tex]\hat{(u\times v)}[/tex]=|w|sin[tex]\theta[/tex][tex]\hat{(u\times w)}[/tex]
    [tex]\hat{(u\times v)}[/tex]=[tex]\hat{(u\times w)}[/tex]
    therefore, v=w
     
  2. jcsd
  3. Apr 7, 2010 #2
    Re: Help!? Linear algebra proof

    I think you're on the right track. One thing to keep in mind is that we can't assume a priori that the two angles are equal, so we're looking at

    [tex]|u||v|\cos\theta_1 = |u||w|\cos\theta_2

    |v|\cos\theta_1 = |w|\cos\theta_2[/tex]

    Similarly,

    [tex]|v|\sin\theta_1 = |w|\sin\theta_2[/tex]

    See what you can do from there
     
  4. Apr 7, 2010 #3
    Re: Help!? Linear algebra proof

    Thanks, i completely forgot bout the angles not being equal.
    however, going with that i can only simplify it down to

    [tex]\upsilon[/tex] [tex]\cdot[/tex] [tex]\upsilon[/tex] = [tex]\omega[/tex] [tex]\cdot[/tex] [tex]\omega[/tex]
     
  5. Apr 7, 2010 #4

    lanedance

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    Homework Helper

    Re: Help!? Linear algebra proof

    so that shows the magnitudes are the same...
     
  6. Apr 7, 2010 #5
    Re: Help!? Linear algebra proof

    yeah, but that only encompasses magnitude not direction.
    anyway i figured it out by using the various laws.

    if, u · v = u · w
    then, u · (v-w) = 0

    if, u x v = u x w
    then, u x (v-w) = 0

    therefore, v-w is both orthogonal and parallel to the non zero vector u, hence v-w = 0
    therefore v=w
     
  7. Apr 7, 2010 #6

    lanedance

    User Avatar
    Homework Helper

    Re: Help!? Linear algebra proof

    once you have the magnitudes, it follows from either

    but yeah thats heaps nicer
     
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