Let's break down the problem step-by-step: --- ## **Step 1: Understand the Question** Given: - **n = 10**, i.e., vectors are in 10-dimensional space. - **s(P, Q)** is the **dot product** of vectors **P** and **Q**. - **L** is a set of non-zero vectors in \(\mathbb{R}^{10}\) such that for every **distinct pair** \(P, Q \in L\), \(s(P, Q) = \). **Goal:** Find the **maximum cardinality** (maximum number of vectors) possible for the set \(L\). --- ## **Step 2: Rephrase Mathematically** We want the largest set of non-zero vectors in \(\mathbb{R}^{10}\) such that **any two distinct vectors are orthogonal**. --- ## **Step 3: Use Linear Algebra Facts** - In an **n-dimensional real vector space**, the **maximum number of mutually orthogonal, non-zero vectors** is **n**. - This is because the set of mutually orthogonal vectors forms an **orthonormal basis**, and the dimension of the space is the maximum number of linearly independent vectors. --- ## **Step 4: Apply to the Problem** - Here, \(n = 10\). - So, the **maximum number of mutually orthogonal non-zero vectors** in \(\mathbb{R}^{10}\) is **10**. --- ## **Step 5: Choose the Answer** The correct answer is: **B. 10** --- ### **Summary Table** | Option | Value | Explanation | |--------|-------|--------------------------------------------------| | A | 9 | Less than possible | | B | 10 | **Maximum possible in 10D space** | | C | 11 | More than possible, not allowed | | D | 100 | More than possible, not allowed | --- ### **Final Answer** \[ \boxed{10} \] The maximum cardinality possible for the set \(L\) is **10**.