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Homework Help: Proving a vector inequality in R^n

  1. Jan 28, 2009 #1

    king vitamin

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    1. The problem statement, all variables and given/known data
    In R^n show that:

    [tex]\Vert\overrightarrow{x - y}\Vert \Vert\overrightarrow{x+y}\Vert \leq \Vert\overrightarrow{x}\Vert^{2} + \Vert\overrightarrow{y}\Vert^{2}[/tex]


    2. Relevant equations

    My main attempts have either been from the Triangle Inequality:
    [tex] \Vert\overrightarrow{x+y}\Vert \leq \Vert\overrightarrow{x}\Vert + \Vert\overrightarrow{y}\Vert[/tex]

    Or from attempts to implement the idea:
    sqrt(a^2 + b^2) <= |a| + |b|

    3. The attempt at a solution
    Everytime I attempt to do this algebraically, the product on the left becomes a distributive catastrophe and I can't get anything sensible out of it, and attempting to square both sides to get rid of the square root just results in the same catastrophe on the RHS. I tried to represent the left hand side as a dot product but that involved cos(theta) which just complicates the problem. If anyone could point me in the right direction it would be appreciated.
     
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  3. Jan 28, 2009 #2

    CompuChip

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    The triangle equality is all you need. Just square both sides and you're almost there already.
     
  4. Jan 28, 2009 #3

    king vitamin

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    I feel more stuck than before! Squaring the Triangle inequality obtains:

    [tex]
    \Vert\overrightarrow{x+y}\Vert^{2} \leq \Vert\overrightarrow{x}\Vert^{2} + \Vert\overrightarrow{y}\Vert^{2} + 2\Vert\overrightarrow{x}\Vert\Vert\overrightarrow{y}\Vert[/tex]

    So do I now need to show that:

    [tex]
    \Vert\overrightarrow{x - y}\Vert \Vert\overrightarrow{x+y}\Vert \leq \Vert\overrightarrow{x+y}\Vert^{2} - 2\Vert\overrightarrow{x}\Vert\Vert\overrightarrow{y}\Vert[/tex]
    ? EDIT: I found a counterexample to this equation, so mark it off the list

    Even more frustrating is this easy result from the Triangle Inequality:

    [tex]
    \Vert\overrightarrow{x+y}\Vert\Vert\overrightarrow{x-y}\Vert \leq \Vert\overrightarrow{x}\Vert^{2} + \Vert\overrightarrow{y}\Vert^{2} + 2\Vert\overrightarrow{x}\Vert\Vert\overrightarrow{y}\Vert[/tex]

    I just can't get rid of the 2xy term!
     
    Last edited: Jan 28, 2009
  5. Jan 28, 2009 #4

    CompuChip

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    Oohhh, now I see, there is a minus sign in one of them.
    Then I think the triangle inequality doesn't help you much... sorry.

    I did get it to work (more or less) through the dot product. You have to think about cos(theta) a bit, but eventually you will be able to use that a - b <= a if b >= 0 and sqrt(a^2 + b^2) <= |a| + |b| to get where you want.
    I hope that's a better hint and you'll get there now.
     
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