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Cross product is cancellative?

  1. Mar 22, 2014 #1
    If u × v = u × w, so v = w ?
  2. jcsd
  3. Mar 22, 2014 #2


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    At first, if two vectors are equal, then they are in the same direction so lets take their magnitudes. We have [itex]uv \sin\alpha=uw\sin\beta [/itex], so you have [itex] v\sin\alpha=w\sin\beta [/itex]. Also [itex] \vec{v} [/itex] and [itex] \vec{w} [/itex] may differ in direction too.
  4. Mar 22, 2014 #3


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    You should have been able to find a simple counterexample. E.g. if u = (1, 0, 0):

    (1, 0, 0) x (x, y, z) = (0, -z, y)

    So, the result only depends on y and z, and x can be anything.
  5. Mar 22, 2014 #4


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    u × v = u × w is equivalent to u × (v-w) =0.
    Therefore u is parallel to v-w, so that v-w is a multiple of u.
    Last edited: Mar 22, 2014
  6. Mar 22, 2014 #5

    D H

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    Not necessarily. What if u is the zero vector?
  7. Mar 23, 2014 #6


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    Quibble. The original question is pointless for u=0.
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