Question

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

• A. $ 4\sqrt{3} $
• B. $ 6\sqrt{3} $
• C. $ 12\sqrt{3} $
• D. $ 18\sqrt{3} $
• E. $ 8\sqrt{3} $

Answer 1

Answer: (C)

$ \because $ $ [\overrightarrow{a}\,\,\overrightarrow{b}\,\,\overrightarrow{c}]=\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c}) $
$ =\overrightarrow{a}.\left( |\overrightarrow{b}|\overrightarrow{c}|\sin \frac{2\pi }{3}\hat{n} \right) $
$ =|\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} $