Question

If \( |\mathbf{a}| = 2\sqrt{2} \) and \( |\mathbf{b}| = 3 \) and the angle between \( \mathbf{a} \) and \( \mathbf{b} \) is \( \frac{\pi}{4} \), and if a parallelogram is constructed with adjacent sides \( \mathbf{p} = 2\mathbf{a} - 3\mathbf{b} \) and \( \mathbf{q} = \mathbf{a} + \mathbf{b} \), then the product of the lengths of both diagonals is:

• A. \( 12\sqrt{26} \)
• B. 6
• C. \( 60\sqrt{2} \)
• D. \( 18\sqrt{260} \)

Hint:

For diagonals in a parallelogram with sides \( \mathbf{p} \) and \( \mathbf{q} \), use the formula \( |\mathbf{p} + \mathbf{q}| \) and \( |\mathbf{p} - \mathbf{q}| \). The magnitudes are then calculated using the properties of vectors and trigonometric identities.

Answer 1

The Correct Option is A