If $\overline{\mathrm{a}}=2\overline{\mathrm{i}}-3\overline{\mathrm{j}}+5\overline{\mathrm{k}}, \overline{\mathrm{b}}=3\overline{\mathrm{i}}-4\overline{\mathrm{j}}+5\overline{\mathrm{k}}$ and $\overline{\mathrm{c}}=5\overline{\mathrm{i}}-3\overline{\mathrm{j}}-2\overline{\mathrm{k}}$, then the volume of the parallelopiped with 

co-terminus edges $\overline{\mathrm{a}}+\overline{\mathrm{b}}, \overline{\mathrm{b}}+\overline{\mathrm{c}} , \overline{\mathrm{c}}+\overline{\mathrm{a}}$ is

$\begin{array}{rcl}\left[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}\right]& =& \left|\begin{array}{ccc}2& -3& 5\\ 3& -4& 5\\ 5& -3& -2\end{array}\right|\\ & =& 2\left(23\right)+3(-31)+5\left(11\right)\\ & =& 46-93+55=8\end{array}$

 $$\begin{gathered} Volume\left(V\right)=[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}] \left|\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 1& 0& 1\end{array}\right|=8\left(1\right(1)-1(-1\left)\right) \\ =16 \end{gathered}$$