Given $p+1$ vectors with pairwise angles $> \pi/2$ in $\mathbb{R}^n$ then every $p$ of them are linearly independent. How to prove this fact?

The motivation for this question is the idea of extending the notion of "linear independence" for points on tangent cones on some metric spaces (Alexandrov spaces), on which we do not have a vector space structure, but have a good notion of angle.