If $\overline{\mathrm{r}}$ is a unit vector satisfying $\overline{\mathrm{r}}\times \overline{\mathrm{a}}=\overline{\mathrm{b}}, \left|\overline{\mathrm{a}}\right|=2 \mathrm{and} \left|\overline{\mathrm{b}}\right|=\sqrt{3}$, then one such $\overline{\mathrm{r}}$=

$\begin{array}{rcl}\left|\overline{\mathrm{r}}\right|& =& 1\\ \overline{\mathrm{r}}\times \overline{\mathrm{a}}& =& \overline{\mathrm{b}} \Rightarrow \overline{\mathrm{a}}\times (\overline{\mathrm{r}}\times \overline{\mathrm{a}})=\overline{\mathrm{a}}\times \overline{\mathrm{b}}\\ & \Rightarrow & (\overline{\mathrm{a}}·\overline{\mathrm{a}}) \overline{\mathrm{r}}-(\overline{\mathrm{a}}·\overline{\mathrm{r}}) \overline{\mathrm{a}}=\overline{\mathrm{a}}\times \overline{\mathrm{b}}\\ |\overline{\mathrm{a}}{|}^2 \overline{\mathrm{r}}-(\overline{\mathrm{a}}·\overline{\mathrm{r}}) \overline{\mathrm{a}}& =& \overline{\mathrm{a}}\times \overline{\mathrm{b}} -\left(1\right)\\ |\overline{\mathrm{r}}\times \overline{\mathrm{a}}|& =& \left|\overline{\mathrm{b}}\right| \Rightarrow 1\times 2\times \mathrm{sin\theta }=\sqrt{3} \Rightarrow \mathrm{\theta }=\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\\ \overline{\mathrm{a}}·\overline{\mathrm{r}}& =& \left|\overline{\mathrm{a}}\right| \left|\overline{\mathrm{r}}\right| \mathrm{cos\theta }=2\times 1\times \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.=1\end{array}$

fromes(1)

$$\begin{gathered} 4\overline{\mathrm{r}}-\overline{\mathrm{a}}=\overline{\mathrm{a}}\times \overline{\mathrm{b}} \\ \overline{\mathrm{r}}=\frac{1}{4}(\overline{\mathrm{a}}+\overline{\mathrm{a}}\times \overline{\mathrm{b}}) \end{gathered}$$