A) \[\frac{\hat{i}+10\hat{j}-18\hat{k}}{5\sqrt{17}}\]
B) \[\frac{\hat{i}-10\hat{j}+18\hat{k}}{5\sqrt{17}}\]
C) \[\frac{\hat{i}-10\hat{j}-18\hat{k}}{5\sqrt{17}}\]
D) \[\frac{\hat{i}+10\hat{j}+18\hat{k}}{5\sqrt{17}}\]
Correct Answer: C
Solution :
\[\vec{A}=2\hat{i}+2\hat{j}-\hat{k}\]and \[\vec{B}=6\hat{i}-3\hat{j}+2\hat{k}\] \[\vec{C}=\vec{A}\times \vec{B}=\left( 2\hat{i}+2\hat{j}-\hat{k} \right)\times \left( 6\hat{i}-3\hat{j}+2\hat{k} \right)\] \[=\left| \,\begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ 2 & 2 & -1 \\ 6 & -3 & 2 \\ \end{matrix}\, \right|\]\[=\hat{i}-10\hat{j}-18\hat{k}\] Unit vector perpendicular to both \[\vec{A}\] and \[\vec{B}\] \[=\frac{\hat{i}-10\hat{j}-18\hat{k}}{\sqrt{{{1}^{2}}+{{10}^{2}}+{{18}^{2}}}\,}\]\[=\frac{\hat{i}-10\hat{j}-18\hat{k}}{5\sqrt{17}}\]
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