The equation of plane passing through the point $\hat{i}+\hat{j}+\hat{k}$ and parallel to the plane $\vec{r}. (2 \hat{i}-\hat{j}+2\hat{k})=5, $ is

$\vec{r}. (2 \hat{i}-\hat{j}+2\hat{k})=3 $

The equation of a plane parallel to the plane $\vec{r}. (2 \hat{i}-\hat{j}+2\hat{k})=5, $ is

$\vec{r}. (2 \hat{i}-\hat{j}+2\hat{k})=d $ ..............(i)

Since it passes through $\hat{i}+\hat{j}+\hat{k}$.

$∴ (\hat{i}+\hat{j}+\hat{k}). (2\hat{i}-\hat{j}+2\hat{k})= d ⇒ 2 -1 + 2 = d ⇒ d = 3.$

Putting d = 3 in (i), we obtain $\vec{r}. (2 \hat{i}-\hat{j}+2\hat{k})=3$.

This is the equation of the required plane.