Find the equation of the plane passing through $(a,b,c)$ and parallel to the plane$\vec r \cdot (\hat i + \hat j + \hat k) = 2.$

.: Any plane parallel to $\vec r \cdot (\hat i + \hat j + \hat k) = 2$
is $\vec r \cdot (\hat i + \hat j + \hat k) = k$ ...(1)

This passes through the point $(a,b,c)$ i.e., $(a\hat i + b\hat j + c\hat k)$

$\therefore$ $(a\hat i + b\hat j + c\hat k) \cdot (\hat i + \hat j + \hat k) = k$

$\Rightarrow$ $(a)(1) + (b)(1) + (c)(1) = k \Rightarrow k = a + b + c$

Putting in (1), the required equation of the plane is $\vec r \cdot (\hat i + \hat j + \hat k) = a + b + c$

$\Rightarrow$ $(x\hat i + y\hat j + z\hat k) \cdot (\hat i + \hat j + \hat k) = a + b + c$

$\Rightarrow$ $x + y + z = a + b + c$