The area of a parallelogram formed from the vectors $\overrightarrow{A}=\hat{i}-2\hat{j}+3\hat{k}$  and $\overrightarrow{B}=3\hat{i}-2\hat{j}+\hat{k}$  as adjacent sides is

$\overrightarrow{A}=\hat{i}-2\hat{j}+3\hat{k}$  and $\overrightarrow{B}=3\hat{i}-2\hat{j}+\hat{k}$  

Area of parallelogram $=\left|\overrightarrow{A}\times \overrightarrow{B}\right|$

Now, $\overrightarrow{A}\times \overrightarrow{B}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -2& 3\\ 3& -2& 1\end{array}\right|=4\hat{i}+8\hat{j}+4\hat{k}$

$\left|\overrightarrow{A}\times \overrightarrow{B}\right|=\sqrt{4^2+8^2+4^2}=\sqrt{96}=4\sqrt{6}units$ .