Lower bound on a spanning set projecting onto any one dimensional subspace

Question

Suppose $\{x_1, \dots, x_n\} \subset \mathbb R^n$ is a spanning set. Let $V$ be an arbitrary one dimensional subspace of $\mathbb R^n$. I am wondering whether there is a constant $c = c(x_1, \dots, x_n)$ independent of $V$, such that $\sum_{j=1}^n \langle P_V x_i, P_V x_i \rangle \ge c$ where $P_V$ denotes the orthogonal projection onto the subspace $V$.