Plane Equations in 3D
A plane in three-dimensional space is a flat, infinite surface. One of the most important facts about a plane is that
it can be described using a normal vector, a vector that points straight “out of” the plane and is perpendicular to every
direction that lies in the plane. If a plane has normal vector \(\mathbf{n}=(a,b,c)\) and passes through a point
\(P_0=(x_0,y_0,z_0)\), then any point \(P=(x,y,z)\) on the plane must satisfy the condition that the displacement
\(\overrightarrow{P_0P}=(x-x_0,\;y-y_0,\;z-z_0)\) is perpendicular to \(\mathbf{n}\). Perpendicular vectors have zero dot product, so the plane is:
\[
\mathbf{n}\cdot(\mathbf{x}-\mathbf{x}_0)=0
\quad\Longrightarrow\quad
a(x-x_0)+b(y-y_0)+c(z-z_0)=0.
\]
This is called the point-normal form. Expanding it gives the widely used general form:
\[
ax+by+cz+d=0
\quad \text{where} \quad
d=-(ax_0+by_0+cz_0).
\]
Many problems become simpler in general form because you can quickly test whether a point lies on the plane by plugging in its coordinates.
The expression \(ax+by+cz+d\) evaluates to 0 on the plane, and its sign tells you which side of the plane the point is on. In addition, the
distance from the origin to the plane is especially neat when you use the normal’s magnitude:
\[
\text{dist}(\mathbf{0},\text{plane})=\frac{|d|}{\sqrt{a^2+b^2+c^2}}.
\]
Another common way to define a plane is with three non-collinear points \(A\), \(B\), and \(C\). “Non-collinear” means they do not all lie on the same line.
These points determine two independent direction vectors inside the plane:
\(\overrightarrow{AB}=B-A\) and \(\overrightarrow{AC}=C-A\). A vector perpendicular to the plane is given by the cross product:
\[
\mathbf{n}=\overrightarrow{AB}\times \overrightarrow{AC}.
\]
If the cross product is the zero vector (or extremely close to it numerically), the points are collinear and there is no unique plane—infinitely many planes contain the same line.
When the cross product is nonzero, it provides a valid normal, and you can use point-normal form with \(P_0=A\) (or any of the three points).
Planes also have useful “secondary” representations. For example, the axis intercepts are found by setting two coordinates to zero in \(ax+by+cz+d=0\).
If \(a\neq 0\), the \(x\)-intercept is \(x=-d/a\) (where the plane crosses the \(x\)-axis). Similar formulas hold for \(y\) and \(z\).
Intercepts help you sketch planes quickly, but they may be undefined if a coefficient is zero (meaning the plane is parallel to that axis).
Finally, the visualization in this calculator draws a finite patch of the plane so you can rotate, pan, and zoom in 3D. This patch is only for display—the true plane
continues infinitely. The drawn normal arrow highlights the plane’s orientation, making it easier to connect the algebraic equation to the geometry (for example, in applications
like “flight-path plane” models, slicing solids, or fitting a best-fit plane to measurement data).
