Question

Let \( \mathbf{a}, \mathbf{b} \) be two unit vectors. If \( \mathbf{c} = \mathbf{a} + 2\mathbf{b} \) and \( \mathbf{d} = 5\mathbf{a} - 4\mathbf{b} \) are perpendicular to each other, find the angle between \( \mathbf{a} \) and \( \mathbf{b} \).

• A. \( \frac{\pi}{6} \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{\pi}{3} \)
• D. \( \frac{\pi}{8} \)

Hint:

For perpendicular vectors \( \mathbf{c} \cdot \mathbf{d} = 0 \), expand and solve for \( \cos \theta \).

Answer 1

The Correct Option is C

Step 1: Condition for Perpendicular Vectors
\[ \mathbf{c} \cdot \mathbf{d} = 0 \] Expanding: \[ (\mathbf{a} + 2\mathbf{b}) \cdot (5\mathbf{a} - 4\mathbf{b}) = 0 \] \[ 5 (\mathbf{a} \cdot \mathbf{a}) - 4 (\mathbf{a} \cdot \mathbf{b}) + 10 (\mathbf{b} \cdot \mathbf{a}) - 8 (\mathbf{b} \cdot \mathbf{b}) = 0 \] \[ 5 - 4\cos \theta + 10\cos \theta - 8 = 0 \] \[ -3 + 6\cos \theta = 0 \] \[ \cos \theta = \frac{1}{2} \] \[ \theta = \frac{\pi}{3} \] Thus, the correct answer is \( \frac{\pi}{3} \).