The equation of a plane passing through the point with position vector $\vec{a}$ and perpendicular to vector $\vec{b}$ is:

Let $\vec{r}$ be the position vector of any arbitrary point $P$ on the plane.The vector $(\vec{r}-\vec{a})$ represents a line segment lying entirely within the plane.The vector $\vec{b}$ is given as perpendicular (normal) to the plane.Since the normal to a plane is perpendicular to every vector lying in the plane:
$$(\vec{r}-\vec{a})\perp \vec{b}$$
Using the dot product property of perpendicular vectors:
$$(\vec{r}-\vec{a})\cdot \vec{b}=0$$