If \(\overline{\mathrm{a}} \cdot\) and \(\overline{\mathrm{c}}\) are unit vectors inclined at \(\frac{\pi}{3}\) with each other and \((\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})) \cdot(\overline{\mathrm{a}} \times \overline{\mathrm{c}})=5\), then the value of \(5[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]=\)

\(\begin{aligned}
|\overline{\mathrm{a}}|=|\overline{\mathrm{c}}| & =1 \\
\overline{\mathrm{a}} \cdot \overline{\mathrm{c}} & =|\overline{\mathrm{a}}||\overline{\mathrm{c}}| \cos \frac{\pi}{3} \\
& =1 \times 1 \times \frac{1}{2} \\
& =\frac{1}{2}
\end{aligned}\)
Now, \([\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})] \cdot(\overline{\mathrm{a}} \times \overline{\mathrm{c}})=5\)
...[Given]
\(\Rightarrow[(\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}) \overline{\mathrm{b}}-(\overline{\mathrm{a}} \cdot \overline{\mathrm{~b}}) \overline{\mathrm{c}}](\overline{\mathrm{a}} \times \overline{\mathrm{c}})=5\)
\(\begin{aligned} & \Rightarrow\left[\frac{1}{2} \overline{\mathrm{~b}}-(\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}) \overline{\mathrm{c}}\right] \cdot(\overline{\mathrm{a}} \times \overline{\mathrm{c}})=5 \\ & \Rightarrow \frac{1}{2} \overline{\mathrm{~b}} \cdot(\overline{\mathrm{a}} \times \overline{\mathrm{c}})-(\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}) \overline{\mathrm{c}} \cdot(\overline{\mathrm{a}} \times \overline{\mathrm{c}})=5 \\ & \Rightarrow \overline{\mathrm{~b}} \cdot(\overline{\mathrm{a}} \times \overline{\mathrm{c}})=10 \\ & \Rightarrow[\overline{\mathrm{~b}} \overline{\mathrm{a}} \overline{\mathrm{c}}]=10 \\ & \Rightarrow-[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]=10 \\ & \Rightarrow[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]=-10 \\ & \Rightarrow 5[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]=-50\end{aligned}\)