20. Plane–Plane Distance

Two planes in ${ℝ}^3$ are either:

• Identical — distance $0$
• Parallel — separated by a constant perpendicular distance
• Intersecting — they cross along a line, distance $0$ at the intersection

20.1. Distance between parallel planes

Two planes are parallel iff their normal vectors are parallel:

are parallel iff $\left({𝐴}_1,{𝐵}_1,{𝐶}_1\right)=𝑘\left({𝐴}_2,{𝐵}_2,{𝐶}_2\right)$ for some $𝑘\ne 0$.

Rewriting both with the same normal vector $\overrightarrow{𝑛}=\left(𝐴,𝐵,𝐶\right)$:

The perpendicular distance between them is:

20.2. Strategy: pick a point on ${𝑃}_1$, measure to ${𝑃}_2$

Equivalent computation: any point ${\overrightarrow{𝑥}}_0$ on ${𝑃}_1$ has point–plane distance to ${𝑃}_2$ equal to the inter-plane distance.

20.3. Non-parallel planes

If the normal vectors are not parallel, the planes intersect along a line in ${ℝ}^3$ — distance $0$. Any pair of non-parallel planes intersects (in 3-D).

The intersection line’s direction is ${\overrightarrow{𝑛}}_1\times {\overrightarrow{𝑛}}_2$ (a vector orthogonal to both normals — see Cross Product).

20.4. Connections

• Planes
• Point–Plane Distance
• Cross Product — direction of intersection line