Question

Let \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) be non-coplanar unit vectors equally inclined to one another at an acute angle \(\theta\). Then \(|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|\) in terms of \(\theta\) is equal to

• A. \((1+\cos\theta)\sqrt{\cos2\theta}\)
• B. \((1+\cos\theta)\sqrt{1-2\cos\theta}\)
• C. \((1-\cos\theta)\sqrt{1+2\cos\theta}\)
• D. None of these

Hint:

Use the Gram determinant for | a·( b× c)|.

Answer 1

The Correct Option is C

For three unit vectors equally inclined with mutual dot product \(\cos\theta\),
\[ |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| = \sqrt{\det \begin{pmatrix} 1 & \cos\theta & \cos\theta \\ \cos\theta & 1 & \cos\theta \\ \cos\theta & \cos\theta & 1 \end{pmatrix}} = (1 - \cos\theta)\sqrt{1 + 2\cos\theta}. \]