The number of elements in the Cartesian product \(A \times B\) is given by the product of the number of elements in set A and the number of elements in set B. So, we have:
\(|\vec{A} \times \vec{B}| = |\vec{A}| \cdot |\vec{B}| = 7\)
Since\(|\vec{A} \times \vec{B}| = 7\), we know that \(p \cdot q = 7\)
To find the value of \(p^2 + q^2\), we need to find the possible values of p and q that satisfy \(p \cdot q = 7\).
The possible pairs of (p, q) that satisfy \(p \cdot q = 7\) are (1, 7) and (7, 1).
For both pairs, \(p^2 + q^2 = 1^2 + 7^2 = 1 + 49 = 50. \)
Therefore, \(p^2 + q^2 = 50. \)
The correct option is (A) 50.