A) \[\left( -\frac{1}{\sqrt{3}} \right)\,\,(\hat{i}+\hat{j}+\hat{k})\]
B) \[\left( \frac{1}{\sqrt{3}} \right)\,\,(\hat{i}-\hat{j}+\hat{k})\]
C) \[\left( \frac{1}{\sqrt{3}} \right)\,\,(\hat{i}+\hat{j}-\hat{k})\]
D) none of these
Correct Answer: A
Solution :
| unit vector \[\bot \] to plane ABC |
| \[=\frac{\overrightarrow{AB}\times \overrightarrow{AC}}{|\overrightarrow{AB}\times \overrightarrow{AC}|}\]\[=\frac{1}{|\overrightarrow{AB}\times \overrightarrow{AC}|}\,\,\left| \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ 1 & 1 & -\,2 \\ 2 & -1 & -1 \\ \end{matrix} \right|\]\[=\frac{\hat{i}\,(-1-2)-\hat{j}\,(-1+4)+\hat{k}\,(-1-2)}{|\overrightarrow{AB}\times \overrightarrow{AC}|}\]\[=\frac{3\hat{i}-3\hat{j}-3\hat{k}}{\sqrt{27}}=\frac{(\hat{i}+\hat{j}+\hat{k})}{\sqrt{3}}\] |
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