Mathematics - Vector Question with Solution | TestHub

$(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})\times \overrightarrow{\mathrm{c}}=\frac{1}{3}|\overrightarrow{\mathrm{b}}||\overline{\mathrm{c}}|\overline{\mathrm{a}}\Rightarrow (\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{c}})\overrightarrow{\mathrm{b}}-(\overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}})\overrightarrow{\mathrm{a}}=\frac{1}{3}|\overline{\mathrm{b}}||\overline{\mathrm{c}}|\overline{\mathrm{a}}$ $\Rightarrow (\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{c}})\overrightarrow{\mathrm{b}}=\left(\frac{1}{3}|\overline{\mathrm{b}}||\overline{\mathrm{c}}|+(\overline{\mathrm{b}}\cdot \overline{\mathrm{c}})\right)\overrightarrow{\mathrm{a}}\Rightarrow \overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{c}}=0$ and $\frac{1}{3}|\overline{\mathrm{b}}||\overline{\mathrm{c}}|+(\overline{\mathrm{b}}\cdot \overline{\mathrm{c}})=0$ $\Rightarrow |\overline{\mathrm{b}}||\overline{\mathrm{c}}|\left(\frac{1}{3}+\cos \theta \right)=0\Rightarrow \cos \theta =-1/3\Rightarrow \sin \theta =\frac{2\sqrt{2}}{3}$