1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

When is the norm of the sum of 2 vectors=sum of norms

  1. Feb 4, 2015 #1
    1. The problem statement, all variables and given/known data
    When is |x+y|=|x|+|y| for arbitrary non-zero vectors x,y∈Rn ie, when does equality hold for the well known inequality |x+y|≤|x|+|y|

    2. Relevant equations
    3. The attempt at a solution
    squaring both sides of the inequality we have
    but (|x+y|)2=∑i(xi+yi)2=∑ixi2+∑iyi2+2∑ixiyi, then, by definition of norm we can cancel out all terms in the inequality but
    2∑ixiyi=<x,y>≤2|x||y| so we see that equality can only hold when both sides of this inequality are equal, therefore we square both sides
    <x,y>2=|x|2|y|2, by definition this expands to ∑ixiyi⋅∑ixiyi=
    but ∑i(xiyi)2=|x|2|y|2
    so our condition for equality of |x+y|=|x|+|y| simplifies to ∑i≠jxixjyiyj=0 or
    this is the answer I arrived at, however I don't feel confident in it for some reason. I found other solutions online for the problem in this exact book that state "when x and y are linearly dependent", however in the newest edition of the book, which I have, the problem actually states : "hint: the answer is not 'they are linearly dependent.'"

    Edit: I've already found some errors in my work, including the fact that I did not expand (|x||y|)^2 properly
    Last edited: Feb 4, 2015
  2. jcsd
  3. Feb 4, 2015 #2
    my ammended solution is ∑i≠jxiyj-xjyi=0
  4. Feb 5, 2015 #3


    User Avatar
    Science Advisor
    Homework Helper

    You are really squaring too much here and drawing wrong conclusions. Think about the dot product. You should get just x.y=|x||y| which you had about halfway through. Then remember that the dot product of two vectors is ##|x||y|cos(\theta)## where ##\theta## is the angle between the two vectors. So?
    Last edited: Feb 5, 2015
  5. Feb 5, 2015 #4
    i see that, however the book explicitly states "the answer is NOT that they are linearly dependent" aka θ=0 which is why I am so confused, the book is "Calculus on Manifolds" by Spivak if you were wondering.
  6. Feb 5, 2015 #5


    Staff: Mentor

    "Linearly dependent" doesn't necessarily mean that θ=0, as you said. If the angle between two vectors were 180°, so that they point in opposite directions, the vectors would be linearly dependent, but it does not follow automatically that |x + y| = |x| + |y|.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Discussions: When is the norm of the sum of 2 vectors=sum of norms
  1. Norm of a vector (Replies: 8)

  2. Is this a norm? (Replies: 13)

  3. Vector norms (Replies: 13)