Question

What is the area of the triangle OAB where O is the origin, $\overrightarrow{OA}=3\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{OB}=2\hat{i}+\hat{j}-3\hat{k}$?

• A. $5\sqrt{6}$ square unit
• B. $\frac{5\sqrt{6}}{2}$ square unit
• C. $\sqrt{6}$ square unit
• D. $\sqrt{30}$ square unit

Answer 1

Answer: (B)

$\frac{5\sqrt{6}}{2}$ square unit

Given in triangle ABC

$\overrightarrow{OA}=3\hat{i}-\hat{j}+\hat{k}$

$\overrightarrow{OB}=2\hat{i}+\hat{j}-3\hat{k}$

area of triangle$=\frac{1}{2}|\overrightarrow{OA}\times \overrightarrow{OB}|$

$\overrightarrow{OA}\times \overrightarrow{OB}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 3& -1& 1\\ 2& 1& -3\end{array}\right|$

$\overrightarrow{OA}\times \overrightarrow{OB}=\hat{i}(3-1)-\hat{j}(-9-2)+\hat{k}(3+2)$

$\overrightarrow{OA}\times \overrightarrow{OB}=2\hat{i}+11\hat{j}+5\hat{k}$

$|\overrightarrow{OA}\times \overrightarrow{OB}|=\sqrt{2^2+{11}^2+5^2}$

$|\overrightarrow{OA}\times \overrightarrow{OB}|=\sqrt{4+121+25}$

$|\overrightarrow{OA}\times \overrightarrow{OB}|=\sqrt{150}$

putting in formula

area of triangle$=\frac{1}{2}|\overrightarrow{OA}\times \overrightarrow{OB}|$

area of triangle$=\frac{1}{2}5\sqrt{6}$

area of triangle$=\frac{5\sqrt{6}}{2}$

Hence option B is correct