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