A) \[4\sqrt{3}\]
B) \[6\sqrt{3}\]
C) \[12\sqrt{3}\]
D) \[18\sqrt{3}\]
Correct Answer: C
Solution :
Key Idea \[[\overrightarrow{\mathbf{a}}\overrightarrow{\mathbf{b}}\overrightarrow{c}]=\overrightarrow{\mathbf{a}}\cdot (\overrightarrow{\mathbf{b}}\times \overrightarrow{\mathbf{c}})=\overrightarrow{\mathbf{a}}(|\overrightarrow{\mathbf{b}}||\overrightarrow{\mathbf{c}}|\sin \theta \widehat{\mathbf{n}})\] Given that,\[|\overrightarrow{\mathbf{a}}|=2,\,\,|\overrightarrow{\mathbf{b}}|=3,\,\,|\overrightarrow{\mathbf{c}}|=4\] \[\therefore \]\[[\overrightarrow{\mathbf{a}}\overrightarrow{\mathbf{b}}\overrightarrow{\mathbf{c}}]=\overrightarrow{\mathbf{a}}\cdot \left( |\overrightarrow{\mathbf{b}}||\overrightarrow{\mathbf{c}}|\sin \frac{2\pi }{3}\widehat{\mathbf{n}} \right)\] \[=|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}||\overrightarrow{\mathbf{c}}|\left( \sin \frac{2\pi }{3} \right)\] \[[\because \,\,\overrightarrow{\mathbf{a}}\cdot \widehat{\mathbf{n}}=|\overrightarrow{\mathbf{a}}||\widehat{\mathbf{n}}|\cos {{0}^{o}}=|\overrightarrow{\mathbf{a}}|]\] \[=2\times 3\times 4\times \frac{\sqrt{3}}{2}\] \[=12\sqrt{3}\]
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