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Vector projection

  1. Sep 20, 2009 #1
    1. The problem statement, all variables and given/known data

    Is it possible for

    projuv=projvu


    2. Relevant equations



    3. The attempt at a solution

    This can only occur if:

    [tex]\frac{|\mathbf{u\cdot v}|}{^{\|u\|^{2}}}\mathbf{u} = \frac{|\mathbf{u\cdot v}|}{^{\|v\|^{2}}}\mathbf{v}[/tex]

    So if either is the zero vector, it is true. How can I prove that it can only be true if either is the zero vector, or v=u?
     
  2. jcsd
  3. Sep 20, 2009 #2

    Office_Shredder

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    You can immediately see from

    That u is a multiple of v (and vice versa). What do you think beyond that?
     
  4. Sep 20, 2009 #3
    That if one is larger/smaller than the other, the projections cannot be equivalent. Is that sufficient to prove it?
     
  5. Sep 20, 2009 #4

    Office_Shredder

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    Do you have a reason for believing that?

    If u is parallel to v, then

    [tex]
    \frac{|\mathbf{u\cdot v}|}{^{\|u\|^{2}}}\mathbf{u} [/tex]

    is parallel to
    [tex] \frac{|\mathbf{u\cdot v}|}{^{\|v\|^{2}}}\mathbf{v}
    [/tex]

    So how can we tell whether they are equal?
     
  6. Sep 20, 2009 #5
    Is it because:

    [tex]\frac{\mathbf{u}}{^{\|u\|}}\frac{1}{\|u\|} = \frac{\mathbf{v}}{^{\|v\|}}\frac{1}{\|v\|}[/tex]

    So we have a constant times a unit vector on each side, therefore the constants must be equal?
     
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