Let $\overrightarrow{\mathrm{r}}$ be a unit vector satisfying  $\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}},\text{ where }\left|\overrightarrow{\mathrm{a}}\right|=\sqrt{3}\text{ and }\left|\overrightarrow{\mathrm{b}}\right|=\sqrt{2}$ then

$\begin{array}{l}\overrightarrow{\mathrm{a}}\times (\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{a}})=\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}\\ 3\overrightarrow{\mathrm{r}}-(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{r}})\overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}\\ \text{ Also }|\overrightarrow{\mathrm{r}}\times \overrightarrow{\mathrm{a}}|=\left|\overrightarrow{\mathrm{b}}\right|\\ \Rightarrow {\sin }^2\mathrm{\theta }=\frac{2}{3}\\ \text{ or }\left(1-{\cos }^2\mathrm{\theta }\right)=\frac{2}{3}\\ \text{ or }\frac{1}{3}={\cos }^2\mathrm{\theta }\\ \Rightarrow \overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{r}}=\pm 1\\ \Rightarrow 3\overrightarrow{\mathrm{r}}\pm \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}\\ \text{ or }\overrightarrow{\mathrm{r}}=\frac{1}{3}(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}\pm \overrightarrow{\mathrm{a}})\end{array}$