The equation of a plane in normal form is $\vec{r} \cdot \hat{n} = d$, where $\hat{n}$ is a unit vector normal to the plane and $d$ is:

In the normal form $\vec{r} \cdot \hat{n} = d$, the constant $d$ is the *perpendicular distance* from the origin to the plane, measured along the unit normal $\hat{n}$. Derivation: for any point $\vec{r}$ on the plane, the projection $\vec{r} \cdot \hat{n}$ is the perpendicular distance from origin to the plane (constant for all points on the plane). Call that constant $d$. If $d > 0$, $\hat{n}$ points from origin toward the plane. If you flip $\hat{n}$, $d$ flips sign. Some textbooks require $d > 0$ by convention.