Question

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

• A. $340$
• B. $\sqrt{25}$
• C. $\sqrt{229}$
• D. $\frac{1}{2}\sqrt{229}$

Answer 1

Answer: (D)

$\vec{a }= 2 \hat{i} -3 \hat{j} +\hat{k}$ and $\vec{b} = 2 \hat{i} +3 \hat{j} + \hat{k}$
$\vec{a} \times\vec{b} = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\ 2&-3&2\\ 2&3&1\end{vmatrix}$
$ = -9 \hat{i} +2\hat{j} + 12\hat{k} $
Area of $\Delta OAB =\frac{1}{2}\left| \vec{a} \times \vec{b}\right| $
$ = \frac{1}{2} \sqrt{81 + 4 + 144}$
$ = \frac{1}{2} \sqrt{229}$