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Math Help - 3-D vector interpolation question

  1. #1
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    Question 3-D vector interpolation question

    I have two phasors of known amplitude and phase (A1, P1 and A2, P2) (shown in red) rotating at the same speed in the Y-Z plane at different locations along the X axis. How can I calculate the amplitude and phase of a third phasor, determined by a (blue) line connecting the two orginal phasors, also rotating in the Y-Z plane located at a known Xo postion (shown in green)?



    Thanks for the help.
    Last edited by castwerks; February 12th 2009 at 04:39 PM.
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  2. #2
    Member Greengoblin's Avatar
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    Since the phasors are rotating at the same speed, and their amplitudes and phases are constant, we can solve it by considering the phasors as being static:

    letting phasor 1 point along the y-axis, it's endpoint is given by the position vector a=x_1i +A_1j+0k where x_1 is the x ordinate of the origin of the first phasor, and i,j, and k are the basis vectors.

    then the endpoint of the second phasor is given by b=\delta xi+A_2\cos(\delta P)j + A_2\sin(\delta P) k. where \delta x =x_2-x_1 is the distance between the origins of the two phasors, and \delta P =P_2-P_1 is the phase diffference between them.

    Then the vector form the enpoint of phasor 1 to the endpoint of phasor 2 (your blue line) is given by:

    r=b-a=(\delta xi+A_2\cos(\delta P)j + A_2\sin(\delta P) k)-(x_1i +A_1j+0k)
    =(\delta x-x_1)i+(A_2\cos(\delta P)-A_1)j+A_2\sin(\delta P)k

    From this you can find the vector equation of the line by using the parameter 0\le t \le 1 :

    r(t)=a+t(b-a)

    =x_1i +A_1j+0k+t[(\delta x-x_1)i+(A_2\cos(\delta P)-A_1)j+A_2\sin(\delta P)k]

    Where r is also a vector.

    Then you can find the equation of the plane passing through x_0 that is parrallel to the y-z plane, and find the intersection of the line with this plane.

    This will give you the endpoint of your third phasor, from which you can easily work out the amplitude and phase.
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