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Solution

Let required unit vector be $\vec{a}=l\hat{i}+m\hat{j}+n\hat{k}$. Equal angles with x and y axis $\Rightarrow l=m$. Orthogonal condition: $\vec{a}\cdot(-\hat{i}+2\hat{j}+2\hat{k})=0$ $-l+2m+2n=0$ Since $l=m$: $-l+2l+2n=0 \Rightarrow l+2n=0 \Rightarrow n=-\dfrac{l}{2}$ So vector $\propto (l,l,-\dfrac{l}{2}) \Rightarrow (2,2,-1)$ Unit vector: $\pm \dfrac{1}{3}(2\hat{i}+2\hat{j}-\hat{k})$