When does the norm of the sum of $n$ vectors equal the sum of the norms of the vectors?

Question

I'm trying to prove the following: Let $V$ be an inner product space over $\mathbb{R}$ or $\mathbb{C}$. For vectors $x_1,\ldots,x_n$, $$\left\lvert\sum\limits_{i=1}^n x_i\right\rvert = \sum\limits_{i=1}^n \lvert x_i \rvert,$$ if and only if for some $j$, there exist nonnegative real numbers $a_i$ such that for all $i \neq j$ we have $x_i=a_ix_j$.

I was able to prove the converse of the statement relatively easily, but I have no clue where to begin for the forward direction.

Answer 1

By the polygon inequality, which is a generalized triangle inequality, we have $$\sum_{i=1}^n |x_i|\geq \left|\sum_{i=1}^n x_i\right|$$ with equality when $x_i$ are all a positive scalar multiples of eachother.