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 :
[A]| Unit vector \[\bot \]to plane ABC \[=\frac{\overrightarrow{AB}\,\,\times \,\,\overrightarrow{AC}}{\left| \overrightarrow{AB}\,\,\times \,\,\overrightarrow{AC} \right|}\] |
| \[\frac{1}{\left| \overrightarrow{AB}\,\,\times \,\,\overrightarrow{AC} \right|}\,\,\,\,\left| \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ 1 & 1 & -\,2 \\ 2 & -\,1 & -\,1 \\ \end{matrix} \right|\] |
| \[=\frac{\hat{i}\,(-\,1-\,2)-\hat{j}\,(-\,1+4)+\hat{k}\,(-\,1-\,2)}{\left| \overrightarrow{AB}\,\,\times \,\,\overrightarrow{AC} \right|}\] |
| \[=\frac{3\hat{i}-3\hat{j}-3\hat{k}}{\sqrt{27}}=\frac{(\hat{i}+\hat{j}+\hat{k})}{\sqrt{3}}\] |
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