A) \[\frac{1}{\sqrt{2}}(\hat{i}+\hat{j})\]
B) \[\frac{1}{\sqrt{5}}(2\hat{i}-\hat{j})\]
C) \[\pm \frac{1}{\sqrt{2}}(\hat{i}-\hat{k})\]
D) None of these
Correct Answer: A
Solution :
Given, two vectors lie in \[xy\] -plane. Therefore, a vector coplanar with them is \[a=x\,\hat{i}+y\,\hat{j}\] \[\therefore \] \[a\bot (\hat{i}-\hat{j})\Rightarrow a.\,(\hat{i}-\hat{j})=0\] \[\Rightarrow \] \[(x\hat{i}+y\hat{i}).\,(\hat{i}-\hat{j})=0\] \[\Rightarrow \] \[x-y=0\] \[\Rightarrow \] \[x=y\] \[\therefore \,a=x\,\hat{i}+x\,\hat{j}\] and \[\left| a \right|=\sqrt{{{x}^{2}}+{{x}^{2}}}=x\sqrt{2}\] \[\therefore \] Required unit vector \[=\frac{a}{\left| a \right|}=\frac{x(\hat{i}+\hat{j})}{x\sqrt{2}}\] \[=\frac{1}{\sqrt{2}}(\hat{i}+\hat{j})\]
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