Question

Find the area of the parallelogram whose diagonals are represented by the vectors $\overrightarrow{a}=2\hat{i}-3\hat{j}+4\hat{k}$ and $\overrightarrow{b}=2\hat{i}-\hat{j}+2\hat{k}.$

Answer 1

$\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& -3& 4\\ 2& -1& 2\end{array}\right|=-2\hat{i}+4\hat{j}+4\hat{k}$
Now, area of the parallelogram $=\frac{|\overrightarrow{a}\times \overrightarrow{b}|}{2}=\frac{\sqrt{4+16+16}}{2}=\frac{6}{2}=3$ sq. units.