Diagonals of a parallelogram are represented by vectors $\overrightarrow{A}=5\hat{i}-4\hat{j}+3\hat{k}\text{and}\overrightarrow{B}=3\hat{i}+2\hat{j}-\hat{k}$ . Area of the parallelogram is :

Area of the parallelogram

$\overline{A}\times \overline{B}\Rightarrow \left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 5& -4& 3\\ 3& 2& -1\end{array}\right|$

$\Rightarrow \hat{i}\left(4-6\right)-\hat{j}\left(-5-9\right)+\hat{k}\left(10+12\right)$

$\Rightarrow \hat{i}\left(-2\right)-\hat{j}\left(-14\right)+22\hat{k}$

$\Rightarrow -2\hat{i}+14\hat{j}+22\hat{k}$

$=-\hat{i}+7\hat{j}+11\hat{k}$

$\left|\overline{A}\times \overline{B}\right|=\sqrt{{\left(1\right)}^2+{\left(7\right)}^2+{\left(11\right)}^2}$

$=\sqrt{1+49+121}$

$=\sqrt{171}$