For two $n$-dimensional real vectors $P$ and $Q$, the operation $s(P, Q)$ is defined as follows: $$s(P, Q) = \sum_{i=1}^n (P[i] \cdot Q[i])$$ 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 $\mathcal{L}$?

The operation $s(P, Q) = \sum_{i=1}^n (P[i] \cdot Q[i])$ is the standard Euclidean dot product (inner product) in $\mathbb{R}^n$.

The condition $s(P, Q) = 0$ for every pair of distinct non-zero vectors $P, Q \in \mathcal{L}$ means that the vectors in $\mathcal{L}$ are mutually orthogonal.

A fundamental theorem of linear algebra states that any set of non-zero, mutually orthogonal vectors is linearly independent.

In an $n$-dimensional real vector space $\mathbb{R}^n$, the maximum number of linearly independent vectors is equal to the dimension of the space, which is $n$.

Here, the vectors are $10$-dimensional ($n = 10$). Therefore, the maximum cardinality possible for the set $\mathcal{L}$ is $10$.

Thus, option (B) is correct.