A) \[\hat{i}-\hat{j}+\hat{k}\]
B) \[\hat{i}+\hat{j}+\hat{k}\]
C) \[\frac{\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}\]
D) \[\frac{\hat{i}-\hat{j}+\hat{k}}{\sqrt{3}}\]
E) none of these
Correct Answer: D
Solution :
\[\because \]\[(\hat{i}+\hat{j})\times (\hat{j}+\hat{k})\] \[=\left| \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ 1 & 1 & 0 \\ 0 & 1 & 1 \\ \end{matrix} \right|\] \[=\hat{i}-\hat{j}+\hat{k}\] and\[|(\hat{i}+\hat{j})\times (\hat{j}+\hat{k})|=\sqrt{1+1+1}=\sqrt{3}\] \[\therefore \]Unit vector perpendicular to both\[\hat{i}+\hat{j}\]and\[\hat{j}+\hat{k}\] \[=\frac{(\hat{i}+\hat{j})\times (\hat{j}+\hat{k})}{|(\hat{i}+\hat{j})\times (\hat{j}+\hat{k})|}\] \[=\frac{\hat{i}-\hat{j}+\hat{k}}{\sqrt{3}}\]
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