A) \[\frac{\pi }{6}\]
B) \[\frac{\pi }{3}\]
C) \[\frac{\pi }{4}\]
D) \[\frac{\pi }{2}\]
Correct Answer: B
Solution :
\[\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\frac{1}{2}\overrightarrow{b}\] \[\Rightarrow \] \[(\overrightarrow{a}.\overrightarrow{c})\overrightarrow{b}-(\overrightarrow{a}.\overrightarrow{b})\overrightarrow{c}=\frac{1}{2}\overrightarrow{b}\] \[\Rightarrow \]\[\left( \overrightarrow{a}.\overrightarrow{c}-\frac{1}{2} \right)\overrightarrow{b}-(\overrightarrow{a}.\overrightarrow{b})\overrightarrow{c}=0\] \[[\because \overrightarrow{b}\ne \overrightarrow{c}]\] \[\Rightarrow \] \[\overrightarrow{a}.\overrightarrow{c}-\frac{1}{2}=0\] and \[\overrightarrow{a}.\overrightarrow{b}=0\] \[\therefore \]Angle between\[\overrightarrow{a}\]and\[\overrightarrow{c}\]is \[\Rightarrow \] \[\overrightarrow{a}.\overrightarrow{c}=\frac{1}{2}\] \[\Rightarrow \] \[|\overrightarrow{a}|.|\overrightarrow{c}|\cos \alpha =\frac{1}{2}\] \[\Rightarrow \] \[\cos \alpha =\frac{1}{2}=\cos \frac{\pi }{3}\] \[\Rightarrow \] \[\alpha =\frac{\pi }{3}\]
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