$\overrightarrow{\mathrm{a}}\text{ and }\overrightarrow{\mathrm{b}}$  are two unit vectors that are mutually perpendicular. A unit vector that is equally inclined to $\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}}\text{ and }\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}\text{ is }$

 Let the required vector $\overrightarrow{r}$ be such that $\overrightarrow{\mathrm{r}}={\mathrm{x}}_1\overrightarrow{\mathrm{a}}+{\mathrm{x}}_2\overrightarrow{\mathrm{b}}+{\mathrm{x}}_3\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}$

We must have $\overrightarrow{\mathrm{r}}\cdot \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{r}}\cdot \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{r}}\cdot (\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})$  (as $\overrightarrow{\mathrm{r}},\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}}\text{ and }\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}$ are unit vectors and $\overrightarrow{r}$ is equally inclined to  $\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}}\text{ and }\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}$)

$\begin{array}{l}\mathrm{now} \overrightarrow{\mathrm{r}}\cdot \overrightarrow{\mathrm{a}}={\mathrm{x}}_1,\overrightarrow{\mathrm{r}}\cdot \overrightarrow{\mathrm{b}}={\mathrm{x}}_2,\overrightarrow{\mathrm{r}}\cdot (\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})={\mathrm{x}}_3\\ \Rightarrow \overrightarrow{\mathrm{r}}=\mathrm{\lambda }(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}\left)\right)\\ \mathrm{also}, \overrightarrow{\mathrm{r}}\cdot \overrightarrow{\mathrm{r}}=1\\ \Rightarrow {\mathrm{\lambda }}^2(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})\cdot (\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}\left)\right)=1\\ \mathrm{or} {\mathrm{\lambda }}^2\left(|\overrightarrow{\mathrm{a}}{|}^2+|\overrightarrow{\mathrm{b}}{|}^2+|\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}{|}^2\right)=1\\ \mathrm{or} {\mathrm{\lambda }}^2=\frac{1}{3}\\ \mathrm{or} \mathrm{\lambda }=\pm \frac{1}{\sqrt{3}}\end{array}$

$\Rightarrow \overrightarrow{\mathrm{r}}=\pm \frac{1}{\sqrt{3}}(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})$