A) \[\frac{5}{2}\,\sqrt{17}\,sq\,units\]
B) \[\frac{2}{5}\,\sqrt{17}\,sq\,unit\]
C) \[\frac{3}{5}\,\sqrt{17}\,sq\,unit\]
D) \[\frac{5}{3}\,\sqrt{17}\,sq\,unit\]
Correct Answer: A
Solution :
| [a] Given \[\overrightarrow{OA}=\,\vec{a}=\,3\hat{i}-6\hat{j}+2\hat{k}\] and |
| \[\overrightarrow{OB}=\,\,\vec{b}=\,2\hat{i}+\hat{j}-2\hat{k}\] |
| \[\therefore \] \[\,(\vec{a}\times \vec{b})\,=\,\left| \,\begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ 3 & -6 & 2 \\ 2 & 1 & -2 \\ \end{matrix}\, \right|\] |
| \[=\,(12-2)\,\hat{i}+(4+6)\hat{j}+\,(3+12)\,\hat{k}\] |
| \[=10\hat{i}+\,10\hat{j}+15\hat{k}\] |
| \[\Rightarrow \] \[|\vec{a}\times \vec{b}|\,\,=\,\sqrt{{{10}^{2}}+\,{{10}^{2}}+{{15}^{2}}}\] |
| \[=\,\sqrt{425}=\,5\sqrt{17}\] |
| Area of \[\Delta OAB=\,\frac{1}{2}\,\,|\vec{a}\times \,\vec{b}|\,=\,\frac{5\sqrt{17}}{2}\,sq.\ unit\] |
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