Mathematics - Vector Question with Solution | TestHub

Area of the parallelogram is given by $\frac{1}{2}\left|\overrightarrow{AC}\times \overrightarrow{BD}\right|$.

$\Rightarrow \mathrm{Area}=\frac{1}{2}\left|\left(-5\hat{i}+\hat{j}-7\hat{k}\right)\times \left(\hat{i}+2\hat{j}+3\hat{k}\right)\right|$

$\Rightarrow \mathrm{Area}=\frac{1}{2}\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ -5& 1& -7\\ 1& 2& 3\end{array}\right|$

$\Rightarrow \mathrm{Area}=\frac{1}{2}\left|17\hat{i}+8\hat{j}-11\hat{k}\right|$

$\Rightarrow \mathrm{Area}=\frac{1}{2}\sqrt{289+64+121}$

$\Rightarrow \mathrm{Area}=\frac{\sqrt{474}}{2}$ square units.