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Homework Help: Scalar triple product coplanarity

  1. Jan 9, 2010 #1
    1. The problem statement, all variables and given/known data

    Show that u, v, w lie in the same plane in R3 if and only if u · (v × w) = 0.

    2. Relevant equations

    3. The attempt at a solution

    if u · (v × w) = 0, then u is orthogonal to vxw, and
    vxw is orthogonal to v and w.

    therefore, u must lie in the same plane determined by v and w (i.e. they are coplanar)

    Is this correct? Also, how would i describe this proof mathematically?

  2. jcsd
  3. Jan 9, 2010 #2
    u · (v × w) = 0

    use geometrical proof

    translate the equation as volume of parallopiped
    the volume is zero only when the height is zero
    or all vectors lie in the same plane
  4. Jan 10, 2010 #3


    User Avatar
    Science Advisor

    Yes, that's correct and, once you have stated exactly HOW you know that two vectors, perpendicular to the same non-zero vector, are co-planar, it IS "mathematical".

    payumooli's suggestion is another way to do it but since you have already done it your own way, I suggest you stay with it.
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