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

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>=Σ_ix_i²
|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(x_i+y_i)²=∑_ix_i²+∑_iy_i²+2∑_ix_iy_i, then, by definition of norm we can cancel out all terms in the inequality but
2∑_ix_iy_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 ∑_ix_iy_i⋅∑_ix_iy_i=
∑_i(x_iy_i)²+∑_i≠jxⁱx^jyⁱy^j
but ∑_i(x_iy_i)²=|x|²|y|²
so our condition for equality of |x+y|=|x|+|y| simplifies to ∑_i≠jxⁱx^jyⁱy^j=0 or
∑_i≠jxⁱx^jyⁱy^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

 

my ammended solution is ∑_i≠jx_iy_j-x_jy_i=0

 

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.