A) \[\frac{1}{\sqrt{11}}(\hat{i}+3\hat{j}-\hat{k})\]
B) \[\frac{1}{\sqrt{11}}(\hat{i}-3\hat{j}+\hat{k})\]
C) \[\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})\]
D) None of these
Correct Answer: A
Solution :
We have, \[r\times a=b\times a\]and \[r\times b=a\times b\] \[\Rightarrow \] \[r\times a=-(r\times b)\] \[\Rightarrow \] \[r\times (a+b)=0\] \[\Rightarrow \] r is parallel to \[a+b\] \[\Rightarrow \] \[r=\lambda (a+b)\] \[\Rightarrow \] \[r=\lambda (\hat{i}+3\hat{j}-\hat{k})\] \[\Rightarrow \]\[|r|=\sqrt{11}\lambda \] \[\therefore \] Required vector \[=\frac{r}{|r|}=\frac{1}{\sqrt{11}}(\hat{i}+3\hat{j}-\hat{k})\]
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