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Coordinate Transformation

  1. Aug 2, 2014 #1
    1. The problem statement, all variables and given/known data
    Prove:
    [tex]\cos\alpha\cdot\cos\alpha'+\cos\beta\cdot\cos\beta'+\cos\gamma\cdot\cos \gamma'=\cos\theta[/tex]
    See drawing Snap1

    2. Relevant equations
    None

    3. The attempt at a solution
    See drawing Snap2. i make the length of the lines 1 and 2 to equal one, for simplicity.
    The projection of line 1 on one of the axes is cos(α).
    ##\cos\alpha\cdot\cos\beta## is the projection of line OA=cos(α) on line 2, which causes line AB to be perpendicular to line 2.
    If i will make the same procedure for all 3 axes and add the 3 projections on line 2 i have to get, see Snap1, line OC which is ##1\cdot\cos\theta##, but i don't know how to do it.
    The book from which i took this problem says i have to solve it from geometrical considerations.
    Where can i find the proof to this problem?
     

    Attached Files:

  2. jcsd
  3. Aug 2, 2014 #2

    ehild

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    Homework Helper
    Gold Member

    You have two straight lines given with their direction cosines, and θ is the angle between the lines?

    The direction cosines are the Cartesian components of the unit tangent vector of the line. ##\vec e_1=<\cos(\alpha);\cos(\beta);\cos(\gamma)> ## and ##\vec e_2=<\cos(\alpha');\cos(\beta');\cos(\gamma')> ##. The dot product of the unit vectors ##\vec e_1## and ##\vec e_2## is ##\vec e_1\cdot\vec e_2 =\cos(\alpha)\cos(\alpha')+\cos(\beta)\cos(\beta')+\cos(\gamma) \cos( \gamma ')##. How is the dot product related to the angle θ between the unit vectors?

    ehild
     
  4. Aug 3, 2014 #3
    Thanks, Ehild, the dot product is just that, the cos between the lines.
     
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