One vertex of a parallelopiped is at the point (1, –1, –2) of rectangular cartesian coordinates. If three adjacent vertices are at (0, 1, 3), (3, 0, –1) and (1, 4, 1) the volume of the parallelopiped is

$\overrightarrow{\mathrm{P}}=\widehat{\mathrm{i}}-\widehat{\mathrm{j}}-2\widehat{\mathrm{k}};\overrightarrow{\mathrm{PS}}=0\widehat{\mathrm{i}}+\widehat{\mathrm{j}}+3\widehat{\mathrm{k}}-(\widehat{\mathrm{i}}-\widehat{\mathrm{j}}-2\widehat{\mathrm{k}})$

$\Rightarrow \overrightarrow{\mathrm{PS}}=\overrightarrow{\mathrm{A}}=-\widehat{\mathrm{i}}+2\widehat{\mathrm{j}}+5\widehat{\mathrm{k}}$

$\overrightarrow{\mathrm{PQ}}=(3\widehat{\mathrm{i}}+0\widehat{\mathrm{j}}-\widehat{\mathrm{k}})-(\widehat{\mathrm{i}}-\widehat{\mathrm{j}}-2\widehat{\mathrm{k}})$

$\Rightarrow \overrightarrow{\mathrm{PR}}=\overrightarrow{\mathrm{B}}=2\widehat{\mathrm{i}}+\widehat{\mathrm{j}}+\widehat{\mathrm{k}}$

$\overrightarrow{PR}=(\widehat{i}+4\widehat{j}+\widehat{k})-(\widehat{i}+\widehat{j}+2\widehat{k})$

$\overrightarrow{\mathrm{PQ}}=\overrightarrow{\mathrm{C}}=5\widehat{\mathrm{j}}+3\widehat{\mathrm{k}}$

volume of parallelopiped $=(\overrightarrow{\mathrm{A}}\times \overrightarrow{\mathrm{B}})\cdot \overrightarrow{\mathrm{C}}$

$\Rightarrow \text{ volume }\left|\begin{array}{ccc}-1& 2& 5\\ 2& 1& 1\\ 0& 5& 3\end{array}\right|$

$\Rightarrow \text{ volume }=(-1)(3-5)-2(6-0)+5(10-0)$

$\Rightarrow \text{ volume }=2-12+50\Rightarrow \text{ volume }=40\text{ unit }$