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Thread: help with 2 questions

  1. #1
    Apr 2008

    help with 2 questions

    i need help with the following question because i cant remember how
    to do them

    1. Determine if the points P(1,1,1), Q(2,3,2), R(3,-2,5) and S(4,0,6)
    are coplanar?
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  2. #2
    Oli is offline
    Apr 2008
    Let A, B, C and D be distinct points in 3 dimensional space.
    Let AB denote the vector from A to B.

    Any three points lie in a plane, so you need only check whether the fourth point lies in the same plane.
    If you take the cross product of ABxAC, then this vector is perpendicular to the plane.
    Now, if D is not on the plane, AD will not be lie in the plane, and so AD will not be perpindicular to ABxAC, and so AD(ABxAC) will be non-zero. Conversely, if it does lie on the plane, AD(ABxAC) will be zero. Therefore:

    AD(ABxAC)=0 if and only if A,B,C and D all lie on the same plane.
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