The units vectors orthogonal to the vector $-\hat i+2\hat j+2\hat k$ and making equal angles with the X and Y axes is(are):

$±\frac{1}{3}(2\hat i+2\hat j-\hat k)$

Let vector be $\vec a = x\hat i+x\hat j+y\hat k$       [∵ It makes equal angles with X and Y axes]

$a.(-\hat i+2\hat j+2\hat k)=0⇒-x+2x+2y= 0⇒x=-2y$

⇒ Also, $x^2+x^2+y^2=1⇒9y^2=1⇒y=±\frac{1}{3}⇒a=±\frac{1}{3}(2\hat i+2\hat j-\hat k)$