Q37

For two $n$-dimensional real vectors $P$ and $Q$, the operation $s(P,Q)$ is defined as follows:

$$s(P,Q) = \displaystyle \sum_{i=1}^n (P[i] \cdot Q[i])$$

Let $\mathcal{L}$ be a set of $10$-dimensional non-zero real vectors such that for every pair of distinct vectors $P,Q \in \mathcal{L}$, $s(P,Q)=0$. What is the maximum cardinality possible for the set $\mathcal{L}$?