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## Homework Statement

When is |x+y|=|x|+|y| for arbitrary non-zero vectors x,y∈R

^{n}ie, when does equality hold for the well known inequality |x+y|≤|x|+|y|

## Homework Equations

|x|

^{2}=<x,x>=Σ

_{i}x

_{i}

^{2}

|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}(x

_{i}+y

_{i})

^{2}=∑

_{i}x

_{i}

^{2}+∑

_{i}y

_{i}

^{2}+2∑

_{i}x

_{i}y

_{i}, then, by definition of norm we can cancel out all terms in the inequality but

2∑

_{i}x

_{i}y

_{i}=<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 ∑

_{i}x

_{i}y

_{i}⋅∑

_{i}x

_{i}y

_{i}=

∑

_{i}(x

_{i}y

_{i})

^{2}+∑

_{i≠j}x

^{i}x

^{j}y

^{i}y

^{j}

but ∑

_{i}(x

_{i}y

_{i})

^{2}=|x|

^{2}|y|

^{2}

so our condition for equality of |x+y|=|x|+|y| simplifies to ∑

_{i≠j}x

^{i}x

^{j}y

^{i}y

^{j}=0 or

∑

_{i≠j}x

^{i}x

^{j}y

^{i}y

^{j}=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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