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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 $\mathcal{L}$?

1. $9$
2. $10$
3. $11$
4. $100$

$S(P, Q)$ is nothing but the dot product of two vectors.

The dot product of two vectors is zero when they are perpendicular, as we are dealing with $10$ dimensional vectors the maximum number of mutually-perpendicular vectors can be $10.$

So option B.
by

edited
The property:

In terms of unit vectors, if $a =\begin{array}{l}a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}\end{array}$

and $\begin{array}{l} b = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \end{array}$ then,

\begin{array}{l}a.b = (a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}).(b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k})\Rightarrow a_1b_1 + a_2b_2 + a_3b_3\end{array}
why cant the vector [0000...0] be included in the set along with other 10 dimensional basis vectors..

@sukesh_reddy in the question it is given to consider NON-ZERO vectors