= Cartesian equation of a plane
{c}
{title2=$\mathbf n\cdot\mathbf x=d$}
= Plane equation
{synonym}
In three-dimensional <Euclidean space>, a nonzero <normal vector> $\mathbf n=(a,b,c)$ determines the <plane equation>
$$
ax+by+cz=d.
$$
For a point $\mathbf a_0$ on the <plane>, $d=\mathbf n\cdot\mathbf a_0$, so the equivalent point-normal form is $\mathbf n\cdot(\mathbf x-\mathbf a_0)=0$. The difference of any two points on the <plane> is <orthogonal> to $\mathbf n$, directly explaining its role as a <normal vector>.
Multiplying $a,b,c,d$ by the same nonzero <scalar> preserves the <set> of solutions. Thus $2x-y+2z=3$ and $-6x+3y-6z=-9$ describe the same <plane>. Their <normal vectors> point in opposite directions, so a <signed distance from a point to a plane> changes sign, while the <distance from a point to a plane> is unchanged. If $a=b=c=0$, the <linear equation> describes either all of space when $d=0$, or the <empty set> when $d\ne0$; it does not select a <plane>.
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