Unbalancing unit vectors

We show that for every $n$ unit vectors $v_1,\dots ,v_n$ in the $d$-dimensional Euclidean space, there exist signs ${\epsilon }_1,\dots ,{\epsilon }_n\in \{\pm 1\}$ such that $\parallel {\epsilon }_1{v}_1+\cdots +{\epsilon }_n{v}_n\parallel \ge \sqrt{2n-d}$, and we characterize the equality cases.