Planes

18. Planes

A plane in $ℝ^{3}$ is the set of points $(𝑥, 𝑦, 𝑧)$ satisfying a single linear equation:

𝐴 𝑥 + 𝐵 𝑦 + 𝐶 𝑧 = 𝐷

The vector $\vec{𝑛} = (𝐴, 𝐵, 𝐶)$ is the normal vector — perpendicular to every vector lying in the plane.

18.1. Point–normal form

If ${\vec{𝑥}}_{0}$ is a known point on the plane and $\vec{𝑛}$ is the normal, then any other point $\vec{𝑥}$ lies on the plane iff $\vec{𝑥} - {\vec{𝑥}}_{0}$ is perpendicular to $\vec{𝑛}$ :

\vec{𝑛} \cdot (\vec{𝑥} - {\vec{𝑥}}_{0}) = 0

Expanding componentwise:

𝑛_{1} (𝑥 - 𝑥_{0}) + 𝑛_{2} (𝑦 - 𝑦_{0}) + 𝑛_{3} (𝑧 - 𝑧_{0}) = 0

Example

Find the plane equation given $\vec{𝑛} = (1, 3, - 2)$ and a point on the plane ${\vec{𝑥}}_{0} = (1, 2, 3)$ .

1 (𝑥 - 1) + 3 (𝑦 - 2) - 2 (𝑧 - 3) = 0

Simplify:

𝑥 - 1 + 3 𝑦 - 6 - 2 𝑧 + 6 = 0 \Rightarrow 𝑥 + 3 𝑦 - 2 𝑧 = 1

18.2. Reading the normal off the equation

In the form $𝐴 𝑥 + 𝐵 𝑦 + 𝐶 𝑧 = 𝐷$ , the coefficients are exactly the components of the normal vector:

\vec{𝑛} = (𝐴, 𝐵, 𝐶)

Example

- 3 𝑥 + \sqrt{2} 𝑦 + 7 𝑧 = 𝜋 \Rightarrow \vec{𝑛} = (- 3, \sqrt{2}, 7)

18.3. See also

Point–Plane Distance — formula and example
Distance Between Planes — parallel and non-parallel cases
Hyperplane — $𝑛$ -dimensional generalization
Cross Product — building a normal from two in-plane vectors