Question

If $\overrightarrow{a}$ is a 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 $\overrightarrow{c}\text{is}\frac{2\pi }{3},\text{then}\left|\right[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\left]\right|$ is equal to

• A. $24$
• B. $12$
• C. $12\sqrt{3}$
• D. $24\sqrt{3}$

Answer 1

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

Given that $|\overrightarrow{a}|=2,|\overrightarrow{b}|=3,|\overrightarrow{c}|=4$ Angle between $\overrightarrow{b}$ and $\overrightarrow{c}\text{is}\frac{2\pi }{3}$ $\therefore \left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=\overrightarrow{a}\cdot (\overrightarrow{b}\times \overrightarrow{c})\Rightarrow |\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]|=\overrightarrow{|a|}\left(|\overrightarrow{b}||\overrightarrow{c}|\sin \frac{2\pi }{3}\widehat{|n|}\right)$
$\because \overrightarrow{a}$ is parallel to $(\overrightarrow{b}\times \overrightarrow{c})$
$=|\overrightarrow{a}||\overrightarrow{b}||\overrightarrow{c}|\left(\sin \frac{2\pi }{3}\right)=2\times 3\times 4\times \frac{\sqrt{3}}{2}$ $=12\sqrt{3}$