A) \[\hat{i}\]
B) \[\frac{\hat{i}+\hat{j}}{\sqrt{2}}\]
C) \[\frac{\hat{i}-\hat{j}}{\sqrt{2}}\]
D) \[\frac{2\hat{i}-\hat{j}}{\sqrt{2}}\]
E) None of these
Correct Answer: E
Solution :
Let the required unit vector be\[\overrightarrow{r}=a\hat{i}+b\hat{j}\]. Then, \[|\overrightarrow{r}|=1\] \[\Rightarrow \] \[\sqrt{{{a}^{2}}+{{b}^{2}}}=1\] \[\Rightarrow \] \[{{a}^{2}}+{{b}^{2}}=1\] ...(i) Since,\[\overrightarrow{r}\]makes an angle of\[45{}^\circ \]with\[\hat{i}+\hat{j}\]and an angle of\[60{}^\circ \]with \[3\hat{i}-4\hat{j},\]therefore \[\cos \frac{\pi }{4}=\frac{\overrightarrow{r}.(\hat{i}+\hat{j})}{|\overrightarrow{r}||\hat{i}+\hat{j}|}\]and\[\cos \frac{\pi }{3}=\frac{\overrightarrow{r}.(3\hat{i}-4\hat{j})}{|\overrightarrow{r}||3\hat{i}-4\hat{j}|}\] \[\Rightarrow \]\[\frac{1}{2}=\frac{a+b}{\sqrt{2}}\]and\[\frac{1}{2}=\frac{3a-4b}{5a}\] \[\Rightarrow \] \[a+b=1\] ?. (ii) And \[3a-4b=\frac{5}{2}\] ?.(iii) There exist no real values of a and b satisfying the Eqs. (i), (ii) and (iii). Hence, no such unit vector exists.
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