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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
    |x|2=<x,x>=Σixi2
    |x+y|≤|x|+|y|
    3. The attempt at a solution
    squaring both sides of the inequality we have
    (|x+y|)2≤|x|2+|y|2+2|x||y|,
    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=
    i(xiyi)2+∑i≠jxixjyiyj
    but ∑i(xiyi)2=|x|2|y|2
    so our condition for equality of |x+y|=|x|+|y| simplifies to ∑i≠jxixjyiyj=0 or
    i≠jxixjyiyj=0
    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

    Dick

    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

    Mark44

    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|.
     
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