A) \[\sqrt{3}\]
B) 2
C) 3
D) \[3\sqrt{3}\]
Correct Answer: C
Solution :
[c] \[\left[ \frac{\overrightarrow{a}\times \overrightarrow{b}}{5}\,\,\,\overrightarrow{b}\times \overrightarrow{c}\,\,\,\,\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{\sqrt{3}} \right]\] \[=\frac{\overrightarrow{a}\times \overrightarrow{b}}{5\sqrt{3}}.(\overrightarrow{b}\times \overrightarrow{c})\times (\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})\] \[=\frac{\overrightarrow{a}\times \overrightarrow{b}}{5\sqrt{3}}.[(\overrightarrow{b}\times \overrightarrow{c})\times \overrightarrow{a}(\overrightarrow{b}\times \overrightarrow{c})\times \overrightarrow{b}+(\overrightarrow{b}\times \overrightarrow{c})\times \overrightarrow{c}]\] \[=\frac{\overrightarrow{a}\times \overrightarrow{b}}{5\sqrt{3}}.[(\overrightarrow{b}.\overrightarrow{a})\times \overrightarrow{c}-(\overrightarrow{c}.\overrightarrow{a})\overrightarrow{b}+(\overrightarrow{b}.\overrightarrow{b})\overrightarrow{c}-(\overrightarrow{c}-\overrightarrow{b})\overrightarrow{b}\] \[+(\overrightarrow{b}.\overrightarrow{c})\overrightarrow{c}-(\overrightarrow{c}.\overrightarrow{c})\overrightarrow{b}]\] \[=\frac{\overrightarrow{a}\times \overrightarrow{b}}{5\sqrt{3}}.[\overrightarrow{c}+4\overrightarrow{c}-9\overrightarrow{b}]=\frac{\overrightarrow{a}\times \overrightarrow{b}}{\sqrt{3}}.\overrightarrow{c}\] Now, \[\overrightarrow{a}.\overrightarrow{b}=1\] \[\Rightarrow \,\,\,\,1\times 2\cos \theta =1\] \[\Rightarrow \,\,\,\,\cos \theta =1/2\] \[\Rightarrow \,\,\,\,\,\sin \theta =\frac{\sqrt{3}}{2}\] \[\Rightarrow \,\,\,\,\overrightarrow{a}\times \overrightarrow{b}=\sqrt{3}\hat{n}\,\,\,\Rightarrow \,\,\,\frac{\overrightarrow{a}\times \overrightarrow{b}}{\sqrt{3}}.\overrightarrow{c}=\frac{\sqrt{3}\hat{n}}{\sqrt{3}}.\overrightarrow{c}=3\]
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