Question

If \( |\mathbf{a}| = 8 \), \( |\mathbf{b}| = 3 \) and \( |\mathbf{a} \times \mathbf{b}| = 12 \), then find the angle between \( \mathbf{a} \) and \( \mathbf{b} \).

• A. \( \frac{\pi}{3} \)
• B. \( \frac{\pi}{6} \)
• C. \( \frac{\pi}{4} \)
• D. None of these

Hint:

For the cross product, use the formula \( |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \sin \theta \) to solve for the angle between the vectors.

Answer 1

The Correct Option is B

The magnitude of the cross product is given by: \[ |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \sin \theta \] Substitute the given values: \[ 12 = 8 \times 3 \times \sin \theta \] \[ \sin \theta = \frac{12}{24} = \frac{1}{2} \] Thus, \( \theta = \sin^{-1} \left( \frac{1}{2} \right) = \frac{\pi}{6} \). Thus, the correct answer is \( \frac{\pi}{6} \).