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Linear combination and orthogonality

  1. Apr 21, 2016 #1
    • Member warned about not using the homework template
    Given the non-zero vectors u, v and w in ℝ3
    Show that there is a non-zero linear combination of u and v that is orthogonal to w.
    u and v must be linearly independant.

    I am not really sure at all. But I have done this:
    This is a screenshot of what I have done. Basicly, I assumed in the end that u and v are not orthogonal, and then I chose some suitable substitution for u and v, and ended up with zero when dotting them with w. Evverything is much appreciated and especially if you have another solution that is better or correct, because I think mine is not that good though.

    https://gyazo.com/707e7c168a1c1a15166fd71be4ae7a81
     
  2. jcsd
  3. Apr 21, 2016 #2

    andrewkirk

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    I'm pretty sure the claim is not correct. Consider the vectors
    u=(5,0,0)
    v=(5,1,0)
    z=(5,0,1)

    The set is linearly independent, but there is no linear combination of u and v that is orthogonal to w.
     
  4. Apr 21, 2016 #3

    Ray Vickson

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    If by z you mean w, then the statement above is false: the linear combination c*u-c*v = (0,-c,0) is orthogonal to w = (5,0,1).

    The result is true, but I will not look at the OP's screenshot (only at a typed version).
     
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