If $\overline{\mathrm{a}} = 2\overline{\mathrm{i}}-\overline{\mathrm{j}}, \overline{\mathrm{b}} = 4\overline{\mathrm{j}} + \overline{\mathrm{k}}, \overline{\mathrm{c}} = 3\overline{\mathrm{k}}$ then$(2\overline{\mathrm{a}} + \overline{\mathrm{b}} + \mathrm{c}) · (-\overline{\mathrm{b}} + 2 \overline{\mathrm{c}}) \mathrm{x} \overline{\mathrm{c}} =$
 

$$\begin{gathered} \left[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}\right] =\left|\begin{array}{ccc}2& -1& 0\\ 0& 4& 1\\ 0& 0& 3\end{array}\right| \\ =2(12-0) \\ =24 \end{gathered}$$

$\begin{array}{rcl} & \Rightarrow & (2\overline{\mathrm{a}} + \overline{\mathrm{b}} + \mathrm{c}) · (-\overline{\mathrm{b}} + 2 \overline{\mathrm{c}}) \mathrm{x} \overline{\mathrm{c}} \\ & =& \left[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}\right] \left|\begin{array}{ccc}2& 1& 1\\ 0& -1& 2\\ 0& 0& 1\end{array}\right|\\ & =& 24\left(2(-1-0)\right)\\ & =& -48\end{array}$