Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Triangle Inequality Proof

  1. Sep 18, 2013 #1
    Hello all,

    I am currently reading about the triangle inequality, from this article

    I am curious, how does the equality transform into an inequality? Does it take on this change because one takes the absolute value of 2uv? Because before the absolute value, 2uv could be a negative value, thus making all of |u|^2 + 2uv + |v|^2 smaller, is this correct?
  2. jcsd
  3. Sep 18, 2013 #2


    User Avatar
    Science Advisor
    Gold Member

    You are correct ... that is why the first inequality appears. The second one is from the Cauchy-Schwartz inequality, as noted.

    These are properties that are required for a metric space.
  4. Sep 18, 2013 #3
    I have one other question. In the article, it says that since both sides of the inequality of non-negative, it is permissible to then square both sides of the inequality. Why would it not be possible to square both sides if both sides were negative?
  5. Sep 18, 2013 #4


    User Avatar
    Science Advisor
    Gold Member

    I'm sure that they said "you can square each term since they are all positive". Try that with this inequality:

    1 - 2 < 1 .... hence the requirement for all positive.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Triangle Inequality Proof
  1. Triangle inequality (Replies: 12)

  2. Inequality Proof (Replies: 3)