Find the area of the triangle formed by the tips of the vectors $\overrightarrow{a}=\widehat{i}-\widehat{j}-3\widehat{k},\overrightarrow{b}=4\widehat{i}-3\widehat{j}+\widehat{k}$ and $\overrightarrow{c}=3\widehat{i}-\widehat{j}+2\widehat{k}$ in sq.units 

$\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{OB}}-\overrightarrow{\mathrm{OA}}=3\widehat{\mathrm{i}}-2\widehat{\mathrm{j}}+4\widehat{\mathrm{k}}$

$\overrightarrow{\mathrm{AC}}=\overrightarrow{\mathrm{OC}}-\overrightarrow{\mathrm{OA}}=2\widehat{\mathrm{i}}+0\widehat{\mathrm{j}}+5\widehat{\mathrm{k}}$

$\text{ Area of triangle }=\frac{1}{2}|\overrightarrow{\mathrm{AB}}\times \overrightarrow{\mathrm{AC}}|$

$\overrightarrow{\mathrm{AB}}\times \overrightarrow{\mathrm{AC}}=\left|\begin{array}{ccc}\widehat{\mathrm{i}}& \widehat{\mathrm{j}}& \widehat{\mathrm{k}}\\ 3& -2& 4\\ 2& 0& 5\end{array}\right|=-10\widehat{\mathrm{i}}-7\widehat{\mathrm{j}}+4\widehat{\mathrm{k}}$

Area of triangle  $=\frac{1}{2}\sqrt{165}=6\cdot 4\mathrm{sq}\cdot \text{ units }$