Question:
Two adjacent sides of a parallelogram PQRS are given by \( \overrightarrow{PQ} = \hat{i} + \hat{j} + \hat{k} \) and \( \overrightarrow{PS} = \hat{i} - \hat{j} \). If the side PS is rotated about the point P by an acute angle \( \alpha \) in the plane of the parallelogram so that it becomes perpendicular to the side PQ, then \( \sin^2 \left( \frac{5\alpha}{2} \right) - \sin^2 \left( \frac{\alpha}{2} \right) \) is equal to:
Two adjacent sides of a parallelogram PQRS are given by \( \overrightarrow{PQ} = \hat{i} + \hat{j} + \hat{k} \) and \( \overrightarrow{PS} = \hat{i} - \hat{j} \). If the side PS is rotated about the point P by an acute angle \( \alpha \) in the plane of the parallelogram so that it becomes perpendicular to the side PQ, then \( \sin^2 \left( \frac{5\alpha}{2} \right) - \sin^2 \left( \frac{\alpha}{2} \right) \) is equal to:
Updated On: Apr 10, 2026
- \( \frac{1}{2} \)
- \( \frac{\sqrt{3}}{2} \)
- \( \frac{\sqrt{3}}{4} \)
- \( \frac{2\sqrt{3}}{5} \)
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The Correct Option is B
Solution and Explanation
Step 1: Set up the vectors.
We are given the two adjacent sides of the parallelogram: \[ \overrightarrow{PQ} = \hat{i} + \hat{j} + \hat{k} \quad \text{and} \quad \overrightarrow{PS} = \hat{i} - \hat{j} \]
Step 2: Apply the rotation.
The rotation of side \( \overrightarrow{PS} \) by an angle \( \alpha \) in the plane results in an angle where the angle between the sides becomes \( 90^\circ \), making the sides perpendicular.
Step 3: Use the trigonometric identity.
The required expression \( \sin^2 \left( \frac{5\alpha}{2} \right) - \sin^2 \left( \frac{\alpha}{2} \right) \) simplifies to \( \frac{\sqrt{3}}{2} \) based on the geometry of the problem.
Final Answer: \( \frac{\sqrt{3}}{2} \)
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