A) \[\frac{(-\hat{j}+\hat{k})}{\sqrt{2}}\]
B) \[\frac{(\hat{k}-\hat{i})}{\sqrt{2}}\]
C) \[\frac{(\hat{i}-\hat{j})}{\sqrt{2}}\]
D) \[\frac{(\hat{i}-\hat{k})}{\sqrt{2}}\]
Correct Answer: A
Solution :
Let\[\overrightarrow{r}=(\hat{i}+\hat{j}+2\hat{k})+\lambda (i+2\hat{j}+\hat{k})\] \[=(1+\lambda )\hat{i}+(1+\lambda )\hat{j}+(2+\lambda )\hat{k}\] ...(i) According to question, \[\overrightarrow{r}.(\hat{i}+\hat{j}+\hat{k})=0\] \[\Rightarrow \] \[(1+\lambda )+(1+2\lambda )+(2+\lambda )=0\] \[\Rightarrow \] \[4\lambda +4=0\] \[\Rightarrow \] \[\lambda =-1\] From Eq. (i), \[\overrightarrow{r}=-\hat{j}+\hat{k}\] \[\therefore \] \[\overrightarrow{r}=\frac{-\hat{j}+\hat{k}}{\sqrt{2}}\]
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