Let the unit vector be $\overrightarrow c=x\hat i+y\hat j+z\hat k$
$\Rightarrow\:x^2+y^2+z^2=1$......(i)
Let the given vectors be $\overrightarrow a=\hat i-\hat j\:\;and\:\:\overrightarrow b=\hat i+2\hat j$
Given that $\overrightarrow c$ is coplanar with $\overrightarrow a\:\:and\:\:\overrightarrow b$ and is $\perp$ to $\overrightarrow a$
$\therefore\:[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]=0\:\:\:and\:\:\:\overrightarrow a.\overrightarrow c=0$
$\Rightarrow\:x-y=0\:\:\:or\:\:\:x=y$....(ii) and
$(\overrightarrow a\times\overrightarrow b).\overrightarrow c=0$
$\overrightarrow a\times\overrightarrow b=\left |\begin {array}{ccc} \hat i &\hat j & \hat k \\1 & -1 & 0 \\1 & 2 & 0\end {array}\right|=3\hat k$
substituting the values of $\overrightarrow a\times \overrightarrow b \:and\:\overrightarrow c$,
$3\hat k.(x\hat i+y\hat j+z\hat k)=0\:\:\Rightarrow\:\:z=0$
Substituting the values of $x,y,z$ in (i) $2y^2=1$ $\Rightarrow\:x=y=\large\frac{1}{\sqrt 2}$
$\therefore \overrightarrow c=\large\frac{1}{\sqrt 2}$$(\hat i+\hat j)$