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

Step 1: Formula for the diagonals of a parallelogram.
In a parallelogram with adjacent sides \( \mathbf{p} \) and \( \mathbf{q} \), the diagonals are given by:
\[ \text{Diagonal 1} = |\mathbf{p} + \mathbf{q}|, \quad \text{Diagonal 2} = |\mathbf{p} - \mathbf{q}| \]

Step 2: Find the expressions for \( \mathbf{p} + \mathbf{q} \) and \( \mathbf{p} - \mathbf{q} \).
We are given:
\[ \mathbf{p} = 2\mathbf{a} - 3\mathbf{b}, \quad \mathbf{q} = \mathbf{a} + \mathbf{b} \]
So, \[ \mathbf{p} + \mathbf{q} = (2\mathbf{a} - 3\mathbf{b}) + (\mathbf{a} + \mathbf{b}) = 3\mathbf{a} - 2\mathbf{b} \]
and \[ \mathbf{p} - \mathbf{q} = (2\mathbf{a} - 3\mathbf{b}) - (\mathbf{a} + \mathbf{b}) = \mathbf{a} - 4\mathbf{b} \]

Step 3: Calculate the magnitudes of the diagonals.
The magnitude of \( \mathbf{p} + \mathbf{q} = 3\mathbf{a} - 2\mathbf{b} \) is:
\[ |\mathbf{p} + \mathbf{q}| = \sqrt{|3\mathbf{a}|^2 + |-2\mathbf{b}|^2 + 2 \cdot (3\mathbf{a}) \cdot (-2\mathbf{b}) \cdot \cos \theta} \]
Substitute \( |\mathbf{a}| = 2\sqrt{2} \), \( |\mathbf{b}| = 3 \), and \( \theta = \frac{\pi}{4} \) into the equation:
\[ |\mathbf{p} + \mathbf{q}| = \sqrt{9(2\sqrt{2})^2 + 4(3)^2 + 2 \cdot 3\mathbf{a} \cdot -2\mathbf{b} \cdot \cos \frac{\pi}{4}} \]

Step 4: Find the value of the product of the diagonals.
Proceed with calculating the result to get: \( 12\sqrt{26} \).

Step 5: Conclusion.
Thus, the product of the lengths of both diagonals is \( 12\sqrt{26} \), corresponding to option (A).