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Triangle Inequality

  1. Sep 21, 2006 #1
    "Use the triangle inequality to prove that:
    [tex]||\vec{a}|| - ||\vec{b}|| \le ||\vec{a} - \vec{b}||[/tex]"

    I can start from that expression and prove it true using the Cauchy-Shwarz inequality but I don't think that's what's asked. Any hints?
  2. jcsd
  3. Sep 21, 2006 #2


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    Do they mean the following triangle inequality:

    [tex]||\vec{a}|| + ||\vec{b}|| \le ||\vec{a} + \vec{b}||[/tex]

    because it can be proved from that by picking a and b correctly.
  4. Sep 21, 2006 #3

    Yeah but it should be ||a|| + ||b|| >= ||a + b||
  5. Sep 21, 2006 #4
    Start with
    [tex]||\vec{u} + \vec{v}|| - ||\vec{v}|| \le ||\vec{u}||[/tex]

    and choose

    [tex]\vec{u} + \vec{v} = \vec{a}[/tex]

    The rest should be pretty self evident after that.
  6. Sep 21, 2006 #5


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    Right, sorry. So you got the answer?
  7. Sep 21, 2006 #6
    I got it but how do I justify saying that?
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