Question

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

• A. 9
• B. 10
• C. 11
• D. 100

Hint:

The maximum number of pairwise orthogonal non-zero vectors in \(\mathbb{R}^n\) is exactly \(n\).

Answer 1

The Correct Option is B

Step 1: Interpret the operation.
The function \(s(P,Q)\) is the standard dot product. The condition \(s(P,Q)=0\) for all distinct \(P,Q \in \mathcal{L}\) means that all vectors in \(\mathcal{L}\) are pairwise orthogonal.

Step 2: Use a linear algebra fact.
In an \(n\)-dimensional real vector space, the maximum number of mutually orthogonal non-zero vectors is \(n\).

Step 3: Apply to the given case.
Here, the dimension is \(10\). Hence, at most \(10\) mutually orthogonal non-zero vectors can exist.

Step 4: Conclusion.
The maximum possible cardinality of \(\mathcal{L}\) is \(10\).