Let the normal to the required plane be $\overrightarrow n$ and a point on the plane be $\overrightarrow a$

Given that the line $\overrightarrow r=2\hat i+\lambda(\hat j-\hat k)$.......(i) lies on the required plane.

$\therefore$ all the points on the line should lie on the plane.

$\therefore (2,0,0)$ is a point on the plane.

$i.e.,$ $ \overrightarrow a=2\hat i$

and since the line lies on the plane,

the line (i) is $\perp$ to the normal to the plane $ (\overrightarrow n)$

Also given that the required plane is $\perp $ to the plane $\overrightarrow r.(\hat i+\hat k)=3$

$i.e.,$ $\overrightarrow n$ is $\perp$ to $ (0,1,-1) \:\:and\:\:(1,0,1)$

$\Rightarrow \overrightarrow n=\left |\begin {array}{ccc} \hat i &\hat j & \hat k\\0 & 1 & -1\\ 1 & 0 & 1\end {array} \right|=(1,-1,-1)$

The equation of the required plane is $(\overrightarrow r-\overrightarrow a).\overrightarrow n=0$

$\therefore$ Eqn. of the required plane is $(\overrightarrow r-2\hat i).(\hat i-\hat j-\hat k)=0$

$\Rightarrow\:\overrightarrow r .(\hat i-\hat j-\hat k)=2$