If $\overline{\mathrm{a}}=\frac{1}{\sqrt{10}}\left(3\overline{\mathrm{i}}+\overline{\mathrm{k}}\right)$ and then the value of $\overline{\mathrm{b}}=\frac{1}{7}(2\overline{\mathrm{i}}+3\overline{\mathrm{j}}-6\overline{\mathrm{k}}),$ then the value of $(2\overline{\mathrm{a}}-\overline{\mathrm{b}}).\left[\right(\overline{\mathrm{a}}\times \overline{\mathrm{b}})\times (\overline{\mathrm{a}}+2\overline{\mathrm{b}}\left)\right]$

$\begin{array}{rcl}\left|\overline{\mathrm{a}}\right|& =& 1 \left|\overline{\mathrm{b}}\right|=1 \Rightarrow \overline{\mathrm{a}}.\overline{\mathrm{b}}=0\\ & =& (2\overline{\mathrm{a}}-\overline{\mathrm{b}}).\left(\left(\overline{\mathrm{a}}\times \overline{\mathrm{b}}\right)\times \left(\overline{\mathrm{a}}+2\overline{\mathrm{b}}\right)\right)\\ & =& (2\overline{\mathrm{a}}-\overline{\mathrm{b}}).\left(\left(\overline{\mathrm{a}}\times \overline{\mathrm{b}}\right)\times \overline{\mathrm{a}}+\left(\overline{\mathrm{a}}\times \overline{\mathrm{b}}\right)\times 2\overline{\mathrm{b}}\right)\\ & =& (2\overline{\mathrm{a}}-\overline{\mathrm{b}}).\left(\overline{\mathrm{b}}-0+2(0-\overline{\mathrm{a}})\right)\\ & =& -{(2\overline{\mathrm{a}}-\overline{\mathrm{b}})}^2=-(4+1-0)=-5\end{array}$