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Are two vectors that are orthogonal to a third parallel?

  1. Jan 19, 2017 #1
    1. The problem statement, all variables and given/known data
    Is it true in three dimensions that any two vectors perpendicular to a third one are parallel to each other?

    2. Relevant equations
    Dot product.

    3. The attempt at a solution
    I've come up with two vectors that were orthogonal to a third and found the angle between them using the definition of the dot product and the angle was not 180 degrees. Therefore I don't think that it's true. I'm really only here to check that I did my math right. Is it actually not true, or do I need to recalculate?
     
  2. jcsd
  3. Jan 19, 2017 #2

    vela

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    You're probably overthinking the problem. You should be able to easily come up with a counterexample which shows the statement is false.
     
  4. Jan 19, 2017 #3
    Not in general, consider the dot products of two non-parallel vectors with the 0 vector.
     
  5. Jan 19, 2017 #4

    StoneTemplePython

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    If you want something visual, you might ponder the right hand rule...
     
  6. Jan 19, 2017 #5
    In my opinion, you should try computing the cross product of two parallel vectors, since the cross product produces a vector normal to both those vectors. Can you do it? If you can answer that question, then you can answer the original question, I think. Given you know how to cross-multiply vectors. But since you're given only the definition of the dot product, you can kindly disregard this post.
     
    Last edited: Jan 19, 2017
  7. Jan 20, 2017 #6

    haruspex

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    XYZ axes?
     
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