$\vec{n}$ is a vector magnitude $2\sqrt{3}$ such that it makes equal angles with coordinate axes. The vector equation of plane passing through (1, -1, 2) and perpendicular to $\vec{n}$ is :

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

The correct answer is Option (1) → $\vec{r}.(\hat{i}+\hat{j}+\hat{k})=2$

$\vec n=2\sqrt{3}(\hat i+\hat j+\hat k)$

makes equal angles with coordinate axes.

let $(\vec a=\hat i-\hat j+2\hat k)$ point on plane

$(\vec r-\vec a).\vec n=0$

$⇒\vec r.\vec n=\vec a.\vec n=2\sqrt{3}$ 

$\vec r(\hat i+\hat j+\hat k)=2\sqrt{3}(1-1+2)$

$\vec r(\hat i+\hat j+\hat k)=2$