Question

If \(|\mathbf{a}| = \sqrt{3}, |\mathbf{b}| = 5, |\mathbf{b}||\mathbf{c}| = 10\), the angle between \(\mathbf{b}\) and \(\mathbf{c}\) is \(\pi/3\), and \(\mathbf{a}\) is perpendicular to \(\mathbf{b} \times \mathbf{c}\), then the value of \(|\mathbf{a} \times (\mathbf{b} \times \mathbf{c})|\) is:

• A. \(20\)
• B. \(30\)
• C. \(60\)
• D. \(40\)

Hint:

The vector triple product involves both magnitude and angle calculations. Use \(\sin\theta\) when calculating the cross-product.

Answer 1

The Correct Option is B

Given: \( |\vec{a}| = \sqrt{3} \), \( |\vec{b}| = 5 \), \( |\vec{c}| = 10 \). The angle between \( \vec{b} \) and \( \vec{c} \) is \( \frac{\pi}{3} \).

Calculate \( \vec{b} \cdot \vec{c} \): \( \vec{b} \cdot \vec{c} = |\vec{b}||\vec{c}| \cos \left(\frac{\pi}{3}\right) = 5 \times |\vec{c}| \times \frac{1}{2} = 10 \).
Also given: \( |\vec{c}| = 4 \).
\( \vec{a} \) is perpendicular to \( \vec{b} \times \vec{c} \).
Therefore, \( \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \).
Calculate \( |\vec{a} \times (\vec{b} \times \vec{c})| \): \( |\vec{a} \times (\vec{b} \times \vec{c})| = |\vec{a}||\vec{b} \times \vec{c}| \sin \frac{\pi}{2} \)
\( = \sqrt{3} \times |\vec{b}||\vec{c}| \sin \frac{\pi}{3} \sin \frac{\pi}{2} \)
\( = \sqrt{3} \times 5 \times 4 \times \frac{\sqrt{3}}{2} \)
\( = 30