A) \[\pm \,3\,(\mathbf{i}+\mathbf{j}+\mathbf{k})\]
B) \[\pm \,\frac{1}{3}\,(\mathbf{i}+\mathbf{j}+\mathbf{k})\]
C) \[\pm \,\frac{1}{\sqrt{3}}\,(\mathbf{i}+\mathbf{j}+\mathbf{k})\]
D) \[\pm \,\sqrt{3}\,(\mathbf{i}+\mathbf{j}+\mathbf{k})\]
Correct Answer: D
Solution :
Let \[\mathbf{r}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}.\] Since \[\mathbf{r}.\mathbf{i}=\mathbf{r}.\mathbf{j}=\mathbf{r}.\mathbf{k}\] \[\Rightarrow x=y=z\] .....(i) Also \[|\mathbf{r}|=\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}=3\Rightarrow x=\pm \sqrt{3}\], {By (i)} Hence the required vector \[\mathbf{r}=\pm \sqrt{3}(\mathbf{i}+\mathbf{j}+\mathbf{k}).\] Trick: As the vector \[\pm \sqrt{3}(\mathbf{i}+\mathbf{j}+\mathbf{k})\] satisfies both the conditions.
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