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

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Homework Statement


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|

Homework Equations


|x|2=<x,x>=Σixi2
|x+y|≤|x|+|y|

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
 
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Answers and Replies

  • #2
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my ammended solution is ∑i≠jxiyj-xjyi=0
 
  • #3
Dick
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my ammended solution is ∑i≠jxiyj-xjyi=0
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?
 
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  • #4
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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?
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.
 
  • #5
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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.
"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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