Let \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) be three vectors such that \(|\overline{\mathrm{a}}|=\sqrt{3}\), \(|\bar{b}|=5, \bar{b} \cdot \bar{c}=10\) and the angle between \(\bar{b}\) and \(\bar{c}\) is \(\frac{\pi}{3}\). If \(\overline{\mathrm{a}}\) is perpendicular to the vector \(\overline{\mathrm{b}} \times \overline{\mathrm{c}}\), then \(|\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})|\) is equal to

\(\begin{aligned} & \overline{\mathrm{b}} \cdot \overline{\mathrm{c}}=10 \\ & \Rightarrow|\overline{\mathrm{b}}||\overline{\mathrm{c}}| \cos \frac{\pi}{3}=10 \\ & \Rightarrow(5)|\overline{\mathrm{c}}|\left(\frac{1}{2}\right)=10 \\ & \Rightarrow|\overline{\mathrm{c}}|=4\end{aligned}\)
\(\begin{aligned} & |\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})|=|\overline{\mathrm{a}}||\overline{\mathrm{b}} \times \overline{\mathrm{c}}| \sin \frac{\pi}{2} \\ & =|\overline{\mathrm{a}}||\overline{\mathrm{b}} \times \overline{\mathrm{c}}| \\ & =|\overline{\mathrm{a}}||\overline{\mathrm{b}}||\overline{\mathrm{c}}| \sin \frac{\pi}{3} \\ & =(\sqrt{3})(5)(4)\left(\frac{\sqrt{3}}{2}\right) \\ & =30\end{aligned}\)