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

Solution

$\therefore$ Area of $\Delta OAB = \frac{1}{2}|\overrightarrow {{\rm{OA}}} \times \overrightarrow {{\rm{OB}}} |$

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

$= \frac{1}{2}\left| {\begin{array}{cccccccccccccccccccc}{\widehat {\rm{i}}}&{\widehat {\rm{j}}}&{\widehat {\rm{k}}}\\2&{ - 3}&2\\2&3&1\end{array}} \right|$

$= \frac{1}{2}\left| {[\widehat {\rm{i}}( - 3 - 6) - \widehat {\rm{j}}(2 - 4) + \widehat {\rm{k}}(6 + 6)]} \right|$

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

$\therefore$ Area of $\Delta OAB = \frac{1}{2}\sqrt {(81 + 4 + 144)} = \frac{1}{2}\sqrt {229}$