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Orthogonal vectors

  1. Jul 14, 2005 #1
    Show that x+y and x-y are orthogonal if and only if x and y have the same norms.

    Can someone get me started?
     
  2. jcsd
  3. Jul 14, 2005 #2

    HallsofIvy

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    1) What does "orthogonal" mean here?

    2) So if x+y and x-y are orthogonal what must be true?

    3) And in order for that to be true what about x and y?
     
  4. Jul 14, 2005 #3

    dextercioby

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    Can you think of a nice geometrical application of this result

    [tex] \left\langle x+y, x-y\right\rangle =0 \Longleftrightarrow ||x||=||y|| [/tex] ?

    Daniel.
     
  5. Jul 14, 2005 #4
    So (x+y)(x-y)=0, which can be turned into ||x||^2 = ||y||^2 take the square root of each side, I get ||x|| = ||y||.

    As for dextercioby's question, if that is true, then x, y, and x+y make up an isosceles right triangle?
     
  6. Jul 14, 2005 #5

    dextercioby

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    The paralelelogramme with perpendicular (onto another) diagonals is a rhombus. Therefore, the vectors have equal modulus. Actually, u've proven the reverse, viz.the geometrical result (theorem/proposition) by algebraic methods only. :wink:

    Daniel.
     
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