The vectors from origin to the points $A$ and $B$ are $\mathbf{a} = 2\hat{i} - 3\hat{j} + 2\hat{k}$ and $\mathbf{b} = 2\hat{i} + 3\hat{j} + \hat{k}$ respectively, then the area of $\triangle OAB$ is equal to

$\frac{1}{2}\sqrt{229}$

The correct answer is Option (4) → $\frac{1}{2}\sqrt{229}$ ##

$∴$ Area of $\triangle OAB = \frac{1}{2} |\vec{OA} \times \vec{OB}|$

$= \frac{1}{2} |(2\hat{i} - 3\hat{j} + 2\hat{k}) \times (2\hat{i} + 3\hat{j} + \hat{k})|$

$= \frac{1}{2} \left| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -3 & 2 \\ 2 & 3 & 1 \end{matrix} \right|$

$= \frac{1}{2} |\hat{i}(-3 - 6) - \hat{j}(2 - 4) + \hat{k}(6 + 6)|$

$= \frac{1}{2} |-9\hat{i} + 2\hat{j} + 12\hat{k}|$

$∴$ Area of $\triangle OAB = \frac{1}{2} \sqrt{81 + 4 + 144} = \frac{1}{2} \sqrt{229}$