The unit vector  orthogonal to $-\stackrel{-}{\mathrm{i}}+2\stackrel{-}{\mathrm{j}}+2\stackrel{-}{\mathrm{k}}$ and  making  equal  angles  with  the  x-axis and y-axis is

$$\begin{gathered} \mathrm{let} \overline{\mathrm{a}}=\mathrm{xi}+\mathrm{yj}+\mathrm{zk}\Rightarrow \left|a\right|\Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2=1 \\ and \mathrm{x}=\mathrm{y} \\ \overline{\mathrm{a}}.\overline{\mathrm{b}}=0 \Rightarrow -\mathrm{x}+2\mathrm{y}+2\mathrm{z}=0 \\ \mathrm{x}+2\mathrm{z}=0 \\ \mathrm{x}=-2\mathrm{z} \\ 4{\mathrm{z}}^2+4{\mathrm{z}}^2+{\mathrm{z}}^2=1 \Rightarrow \mathrm{z}=\pm \frac{1}{3} \\ \overline{\mathrm{a}}=\pm \frac{1}{3} (2\mathrm{i}+2\mathrm{j}+\mathrm{k}) \end{gathered}$$