Question

Let \( \overrightarrow{PS} = \hat{i} + \hat{j} \) and \( \overrightarrow{PQ} = -\hat{j} + \hat{k} \). If \( \overrightarrow{PS} \) must be rotated by an angle \( \alpha \) such that \( \overrightarrow{PS} \) is perpendicular to \( \overrightarrow{PQ} \), then \( \left( \sin^2 \frac{5\alpha}{2} - \sin^2 \frac{\alpha}{2} \right) \) equals

• A. \( \frac{1}{2} \)
• B. \( 1 \)
• C. \( \frac{\sqrt{3}}{2} \)
• D. \( 0 \)

Hint:

For angle-based vector questions, first find the original angle using the dot product formula. Then compare it with the required angle and substitute the rotation angle into the given trigonometric expression.

Answer 1

The Correct Option is C

Step 1: Determine the angle between the given vectors.
The vectors are given as: \[ \overrightarrow{PS} = \hat{i} + \hat{j} \] and \[ \overrightarrow{PQ} = -\hat{j} + \hat{k}. \] Let the angle between these vectors be \( \theta \). Using the dot product formula: \[ \cos \theta = \frac{\overrightarrow{PS} \cdot \overrightarrow{PQ}} {|\overrightarrow{PS}| \, |\overrightarrow{PQ}|} \] First, compute the dot product: \[ (\hat{i} + \hat{j}) \cdot (-\hat{j} + \hat{k}) = 0 - 1 + 0 = -1. \] Next, find the magnitudes of the vectors: \[ |\overrightarrow{PS}| = \sqrt{1^2 + 1^2} = \sqrt{2}, \] \[ |\overrightarrow{PQ}| = \sqrt{(-1)^2 + 1^2} = \sqrt{2}. \] Therefore, \[ \cos \theta = \frac{-1}{\sqrt{2} \times \sqrt{2}} = -\frac{1}{2}. \] Hence, \[ \theta = 120^\circ. \]
Step 2: Calculate the minimum angle of rotation \( \alpha \).
The initial angle between the vectors is \( 120^\circ \). To make \( \overrightarrow{PS} \) perpendicular to \( \overrightarrow{PQ} \), the angle between them must become \( 90^\circ \). Thus, the minimum rotation required is: \[ \alpha = 120^\circ - 90^\circ = 30^\circ. \]
Step 3: Evaluate the given trigonometric expression.
We need to find: \[ \sin^2 \frac{5\alpha}{2} - \sin^2 \frac{\alpha}{2}. \] Substituting \( \alpha = 30^\circ \): \[ \sin^2 \frac{5 \times 30^\circ}{2} - \sin^2 \frac{30^\circ}{2} = \sin^2 75^\circ - \sin^2 15^\circ. \] Using the identity: \[ \sin^2 A - \sin^2 B = \sin(A + B)\sin(A - B), \] we get: \[ \sin^2 75^\circ - \sin^2 15^\circ = \sin(90^\circ)\sin(60^\circ). \] \[ = 1 \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}. \]
Step 4: Conclusion.
Therefore, the required value of the expression is:
Final Answer: \( \dfrac{\sqrt{3}}{2} \).