Cross two edge vectors to get the normal, then substitute a point. For (1,0,0), (0,1,0) and (0,0,1) the normal is (1, 1, 1) and the plane is x + y + z = 1. The coefficients of a plane equation always ARE the normal vector, which is the fact everything else here follows from.
Three points, one plane
Two points fix a line. Three fix a plane — provided they are not already on a line themselves.
Physically it is the reason a three-legged stool never rocks and a four-legged one sometimes does. Three feet always lie in a plane; a fourth has to be lucky.
The construction is direct: take two vectors along the plane, from the first point to each of the others, and cross them. The cross product is perpendicular to both, so it is perpendicular to every direction in the plane — which is what a normal vector is.
The normal is the equation
In ax + by + cz = d, the numbers a, b and c are the components of the normal vector. Not related to it — they are it.
That makes the equation readable rather than arbitrary. It says the dot product of the position with the normal is a constant, and a dot product measures how far along a direction something lies. So the plane is the set of points that sit the same distance along the normal.
Everything else follows. The distance formula divides by the normal’s length in order to measure in units of distance rather than units of the normal. The angle between two planes is the angle between their normals. Two planes are parallel exactly when their normals are.
One consequence worth stating: the equation is only fixed up to scale. Multiplying every coefficient by the same non-zero number describes the same plane, so two correct answers to the same problem can look different.
Signed distance
The distance from a point to a plane is measured along the normal — the shortest route, and the only one perpendicular to the surface.
Substituting the point into the equation gives a number that is not the distance yet; dividing by the normal’s length makes it one.
Before that absolute value, the quantity is signed, and the sign is the useful part: positive means the point is on the side the normal points to, negative means the other side, and zero means it lies in the plane.
That is exactly what a clipping test, a collision check or a shadow calculation needs. “How far” is often less important than “which side”, and a page that reports only the absolute distance has thrown the answer away.
A signed distance of zero for a fourth point is also the coplanarity test: all four points lie in one plane.
When three points are not enough
Three points on a straight line do not fix a plane. Infinitely many planes contain a single line — imagine a page rotating about its spine.
The arithmetic says so cleanly: the two edge vectors are parallel, so their cross product is the zero vector, and there is no direction to call the normal.
This is the reason the cross product here is computed exactly. Three nearly collinear points give a very small normal, and normalising it amplifies whatever rounding error it contains into a plane tilted at an arbitrary angle. An exact computation returns a genuine zero and the page refuses, which is the honest answer.
Where it gets used
Computer graphics. Every surface is a mesh of triangles, and every triangle carries a normal. Lighting is a dot product with it, and which side you are looking at is the sign of that product.
Collision and clipping. Signed distance to a plane decides inside from outside, and a convex shape is an intersection of half-spaces defined exactly this way.
Surveying and construction. Fitting a plane to measured points to check whether a surface is flat, and measuring the deviations from it.
Data analysis. A plane of best fit through points in three dimensions is the three-dimensional version of a line of best fit, and principal component analysis finds it.
Sources and methodology
The vector formulation is standard; these are the references.
Method. The normal is a cross product on exact rationals, and that exactness is doing real work: an all-zero normal is precisely what tells you the three points were collinear, and a floating-point cross product of three nearly-collinear points gives a small non-zero vector rather than a zero one, which would produce a plane out of rounding noise. The distance is reported signed as well as absolute, because the sign carries which side — the part a clipping or collision test needs. The suite checks the normal is perpendicular to both edge vectors on six hundred generated triples. That engine is verified on every change against 119 hand-written assertions, including that the computed normal has a dot product of zero with both edge vectors on every generated point triple, and that a point lying in the plane is reported at distance exactly zero. The count and the per-case breakdown are published on the formula verification page.
Read the guide
The cross product this rests on — what it measures and when it vanishes — is worked through on the Vector Calculator.