Question

If $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$ and $\overrightarrow{c},$ $\left|\overrightarrow{a}\right|=2,\left|\overrightarrow{b}\right|=3,\left|\overrightarrow{c}\right|=4$ and the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is $\frac{2\pi }{3}$, then $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$ is equal to

• A. $4\sqrt{3}$
• B. $6\sqrt{3}$
• C. $12\sqrt{3}$
• D. $18\sqrt{3}$

Answer 1

The correct option is C $12\sqrt{3}$

Given that $\left|\overrightarrow{a}\right|=2,\left|\overrightarrow{b}\right|=3,\left|\overrightarrow{c}\right|=4$

$\therefore \left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})$

$=\overrightarrow{a}\left(\left|\overrightarrow{b}\right|\left|\overrightarrow{c}\right|\sin \frac{2\pi }{3}\hat{n}\right)$

$=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\left|\overrightarrow{c}\right|\left(\sin \frac{2\pi }{3}\right)=2\times 3\times 4\times \frac{\sqrt{3}}{2}$

$=12\sqrt{3}$