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Help with a basic linear algebra proof

  1. Feb 12, 2009 #1
    1. The problem statement, all variables and given/known data

    Let x and y be nonzero vectors in Rn. Prove ||x+y|| = ||x|| + ||y|| if and only if y = cx for some c > 0.

    2. Relevant equations

    Formula for vector magnitude, basic properties of vectors, possibly other vector formulas

    3. The attempt at a solution

    I have proved the first part, that is, that ||x+y|| = ||x|| + ||y|| if y=cx for some c > 0.

    Now I have to prove the second part:
    If ||x+y|| = ||x|| + ||y|| there is some c > 0 such that y = cx.

    I don't know how to begin proving this. I'm guessing I need to find c using the equation. However I can't think of a method that allows me to do this.

    If someone could give me hint or help me get started, I would greatly appreciate it. Thanks.
  2. jcsd
  3. Feb 12, 2009 #2


    Staff: Mentor

    Isn't ||x + y||^2 = (x + y) . (x + y)?
  4. Feb 12, 2009 #3
    Thank you, that was enough for me to figure it out :)
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