Mathematics - Vector Question with Solution | TestHub

Given,

$\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c} )=\frac{\sqrt{3}}{2} (\overrightarrow{b}+\overrightarrow{c} )$

$\Rightarrow \left(\overrightarrow{a}·\overrightarrow{c}\right)\overrightarrow{b}-\left(\overrightarrow{a}·\overrightarrow{b}\right)\overrightarrow{c}=\frac{\sqrt{3}}{2} \overrightarrow{b}+\frac{\sqrt{3}}{2}\overrightarrow{c}$

On comparing, we get

$-\left(\overrightarrow{a}·\overrightarrow{b}\right)=\frac{\sqrt{3}}{2}$

$\Rightarrow \left(\overrightarrow{a}·\overrightarrow{b}\right)=-\frac{\sqrt{3}}{2}$

$\Rightarrow \left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta =-\frac{\sqrt{3}}{2}$

Given $\overrightarrow{a},\overrightarrow{b}$ are unit vector,

$\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=1$

From equation  $\left(i\right)$ 

$\cos \theta =-\frac{\sqrt{3}}{2}$ 

$\Rightarrow \cos \theta =\cos \frac{5\mathrm{\pi }}{6}$

$\Rightarrow \theta =\frac{5\mathrm{\pi }}{6}$