The area of a triangle whose vertices are  $A(1,-1,2)$ , $B(2,1,-1)\text{ and }C(3,-1,2)\text{ is: }$

$\overrightarrow{AB}=(2\mathbf{i}+\mathbf{j}-\mathbf{k})-(\mathbf{i}-\mathbf{j}+2\mathbf{k})$

$\begin{array}{l}=\mathbf{i}+2\mathbf{j}-3\mathbf{k}\\ \overline{AC}=(3\mathbf{i}-\mathbf{j}+2\mathbf{k})-(\mathbf{i}-\mathbf{j}+2\mathbf{k})=2\mathbf{i}\end{array}$

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

$\begin{array}{l}=\frac{1}{2}\left|\right(\mathbf{i}+2\mathbf{j}-3\mathbf{k})\times 2\mathbf{i}|\\ =\frac{1}{2}|-4\mathbf{k}-6\mathbf{j}|=|-3\mathbf{j}-2\mathbf{k}|\\ =\sqrt{13}\end{array}$