3D Geometry

Equations of Planes

Connect the geometry of a plane to vector, scalar-product and Cartesian equations.

Scalar-product form

If a plane has normal vector \(\mathbf n=\begin{pmatrix}a\b\c\end{pmatrix}\) and contains point \(P(x_0,y_0,z_0)\), then

\[\mathbf n\cdot\begin{pmatrix}x-x_0\y-y_0\z-z_0\end{pmatrix}=0.\]
nP(x,y,z)plane
A normal vector is perpendicular to every direction lying in the plane.

Cartesian form

This expands to

\[ax+by+cz=d.\]

The coefficients of \(x,y,z\) give the normal vector \(\mathbf n=(a,b,c)\).