Question

Find the area of the triangle formed by O, A, B when $\overrightarrow{OA}=\hat{i}+2\hat{j}+3\hat{k},\overrightarrow{OB}=-3\hat{i}-2\hat{j}+\hat{k}.$

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ \overrightarrow{OA}=\hat{i}+2\hat{j}+3\hat{k} \\ \overrightarrow{OB}=-3\hat{i}-2\hat{j}+\hat{k} \\ \overrightarrow{OA}\times \overrightarrow{OB}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 2& 3\\ -3& -2& 1\end{array}\right| \\ =8\hat{i}-10\hat{j}+4\hat{k} \\ \left|\overrightarrow{OA}\times \overrightarrow{OB}\right|=\sqrt{64+100+16} \\ =\sqrt{180} \\ =6\sqrt{5} \\ \text{Area of the triangle=}\frac{1}{2}\left|\overrightarrow{OA}\times \overrightarrow{OB}\right| \\ \text{=}\frac{1}{2}\left(6\sqrt{5}\right) \\ \text{=3}\sqrt{5}\text{sq. units} \end{gathered}$$