Question

Let \( (\cdot, \cdot) \) denote the standard inner product on \( {R}^n \). Let \( V = \{v_1, v_2, v_3, v_4, v_5\} \subset {R}^n \) be a set of unit vectors such that \( (v_i, v_j) \) is a non-positive integer for all \( 1 \leq i \neq j \leq 5 \). Define \( N(V) \) to be the number of pairs \( (r, s) \), \( 1 \leq r, s \leq 5 \), such that \( (v_r, v_s) \neq 0 \). The maximum possible value of \( N(V) \) is equal to:

• A. 9
• B. 10
• C. 14
• D. 5

Hint:

For problems involving inner products, analyze the orthogonality and constraints on vector relations.

Answer 1

The Correct Option is A

Step 1: Understanding the inner product condition. The condition \( (v_i, v_j) \neq 0 \) means that the vectors are not orthogonal. Since the vectors are unit vectors and \( (v_i, v_j) \) is a non-positive integer, the structure of \( V \) determines the maximum \( N(V) \). Step 2: Maximizing \( N(V) \). Considering the constraints, the maximum number of non-zero inner products is \( N(V) = 9 \). Step 3: Conclusion. The maximum possible value of \( N(V) \) is \( {(1)} 9 \).