Plane

A flat 2D sheet in space, pinned down by one point on it plus a normal vector.

A plane in space is fixed by one point on it, plus one vector pointing straight off the plane — its normal vector — since that single perpendicular direction already rules out every tilt the plane could have except the one it actually has. Its equation is $a(x-x_0)+b(y-y_0)+c(z-z_0)=0$ , or $ax+by+cz=d$ , where $\mathbf{n}=\langle a,b,c\rangle$ is a normal vector and $(x_0,y_0,z_0)$ is the known point. When three non-collinear points are given instead, the cross product of two vectors built between them is automatically perpendicular to the plane, giving a normal vector directly.

Examples

The plane through $P_1=(1,0,1)$ , $P_2=(2,1,0)$ , $P_3=(-2,2,2)$ has normal $\mathbf{n}=\langle 3,2,5\rangle$ and equation $3x+2y+5z=8$ .
Every flat face of a CNC-machined part is stored as a plane equation with an outward-facing normal vector.