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Homework Help: Vector Problem

  1. Aug 14, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove [tex]\mathbf{PQ}\cdot \mathbf v=\int_a^b\frac{\textup d\gamma}{\textup d t}(t)\cdot\mathbf v\textup d t[/tex]
    where [tex] \mathbf P=\gamma(a)[/tex] and [tex]\mathbf Q=\gamma(b)[/tex]




    3. The attempt at a solution
    I get
    [tex]
    \int_a^b\frac{\textup d\gamma}{\textup d t}(t)\cdot\mathbf v\textup d t=v_x\int_a^b{\frac{\textup d\gamma_x}{\textup d t}\textup d t}+v_y\int_a^b{\frac{\textup d\gamma_y}{\textup d t}\textup d t}+v_z\int_a^b{\frac{\textup d\gamma_z}{\textup d t}\textup d t}\\
    =v_x(\gamma_x(b)-\gamma_x(a))+v_y(\gamma_y(b)-\gamma_y(a))+v_y(\gamma_y(b)-\gamma_y(a))\\
    =\mathbf v\cdot(\mathbf{Q-P})[/tex]

    I'm not sure how to turn this into what is given. I'm not even sure I know what the left hand side of the given identity means. Is it the same thing as
    [tex]\mathbf{P}(\mathbf Q}\cdot \mathbf v)[/tex]
    Any help would be appreciated. Thanks in advance :)
     
  2. jcsd
  3. Aug 14, 2010 #2
    By the way, [tex]v[/tex] is assumed to be a unit vector.
     
  4. Aug 14, 2010 #3
    hopefully somebody with more authority can correct me but...

    I'm very certain that PQ is defined as (Q - P) so you are correct (didn't look through the whole derivation though)
     
  5. Aug 14, 2010 #4
    Oh, I get it. It's the vector from P to Q. Thanks heaps for that. I was so busy trying to figure out where I'd gone wrong that it never occurred to me that I may have it right. :)
     
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