Given vectors $v_1, v_2 ... v_n \in \mathbb{R}^n$ satisfying $||v_i||_2 = 1$ for all $i$ and $0 \leq\langle\ v_i,v_j\rangle \leq 1$ for all $i, j$. (Not all of them are 0 or 1). Does there always exist a rotation matrix which when applied to these vectors bring them to the positive orthant? Clearly the fact is true for n = 2 and 3. Does it generalize to higher dimensions?
FYI, Not a HW Problem.
