The unit vector in the ZOX plane making angles 45° and 60° respectively, with $\vec a = 2\hat i+2\hat j-\hat k$ and $\vec b=\hat j-\hat k$, is

$\frac{1}{\sqrt{2}}(\hat i-\hat k)$

Let the required vector be $\vec r = x\hat i+y\hat k$

Since $\vec r$ is a unit vector.

$∴x^2 + y^2=1$   ...(i)

It is given that $\vec r$ makes 45° and 60° angles with $\vec a$ and $\vec b$ respectively.

$∴os 45° =\frac{\vec r.\vec a}{|\vec r||\vec a|}$ and $\cos 60° =\frac{\vec r.\vec b}{|\vec r||\vec b|}$

$⇒\frac{1}{\sqrt{2}}=\frac{2x-y}{3}$ and $\frac{1}{2}=-\frac{y}{\sqrt{2}}$

$⇒2x -y=\frac{3}{\sqrt{2}}$ and $y=-\frac{1}{\sqrt{2}}⇒x=\frac{1}{\sqrt{2}},y-\frac{1}{\sqrt{2}}$

Hence, $\vec r=\frac{1}{\sqrt{2}}(\hat i-\hat k)$