Let \(\overline{\mathrm{a}}, \overline{\mathrm{b}}\) and \(\overline{\mathrm{c}}\) be three non-zero vectors such that no two of them are collinear and \((\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times \overline{\mathrm{c}}=\frac{1}{3}|\overline{\mathrm{~b}}||\overline{\mathrm{c}}| \overline{\mathrm{a}}\). If \(\theta\) is the angle between vectors \(\overline{\mathrm{b}}\) and \(\overline{\mathrm{c}}\), then the value of \(\operatorname{cosec} \theta\) is

Given: \((\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times \overline{\mathrm{c}}=\frac{1}{3}|\overline{\mathrm{~b}}||\overline{\mathrm{c}}| \overrightarrow{\mathrm{a}}\)
We know that,
\((\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \dot{\times} \overline{\mathrm{c}}=(\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}) \overline{\mathrm{b}}-(\overline{\mathrm{b}} \cdot \overline{\mathrm{c}})^{\bar{a}}\)
- On comparing, we get
\(\begin{aligned}
& \frac{1}{3}|\overline{\mathrm{~b}}||\overline{\mathrm{c}}|=-\overline{\mathrm{b}} \cdot \overline{\mathrm{c}} \\
& \Rightarrow \frac{1}{3}|\overline{\mathrm{~b}}||\overline{\mathrm{c}}|=-|\overline{\mathrm{b}}||\overline{\mathrm{c}}| \cos \theta \\
& \Rightarrow \cos \theta=\frac{-1}{3} \\
& \Rightarrow \cos ^2 \theta=\frac{1}{9} \\
& \begin{aligned}
\sin ^2 \theta & =1-\cos ^2 \theta \\
\quad= & 1-\frac{1}{9}
\end{aligned}
\end{aligned}\)
\(\begin{array}{ll}\therefore \quad & \sin ^2 \theta=\frac{8}{9} \\ \therefore \quad & \sin \theta=\sqrt{\frac{8}{9}}=\frac{2 \sqrt{2}}{3} \\ & \operatorname{cosec} \theta=\frac{3}{2 \sqrt{2}}\end{array}\)