Planes in Space

Definition

A plane in space is a two-dimensional surface defined by a point $P_0$ and a normal vector $\mathbf{n}$ that is perpendicular to every vector lying within the plane.

Why It Matters

We live in 3D, but we build on planes. Every floor, wall, and screen is a 2D constraint in 3D space. If you don’t understand the normal vector, you can’t calculate how forces (like wind or weight) hit a surface. This math is the “language of surfaces” for engineers, architects, and animators. It is the first step in moving from “points” to “structures.”

Core Concepts

• Vector Equation: $\mathbf{n} \cdot \vec{P_0P} = 0$, expressing that any vector in the plane must be orthogonal to the normal vector.
  • How to read: “The dot product of the normal vector n and the vector from P zero to P is equal to zero.”
  • Meaning: Every in-plane vector is perpendicular to the normal $\mathbf{n}$.
• Standard Form: $Ax + By + Cz = D$, where the coefficients $\langle A, B, C \rangle$ represent the components of the normal vector.
  • How to read: “The constant A times x plus the constant B times y plus the constant C times z is equal to the constant D.”
  • Meaning: Linear equation in 3D—coefficients directly give the normal vector.
• Distance from Point to Plane: The distance $d$ from a point $S$ to a plane through $P$ with normal $\mathbf{n}$ is the magnitude of the scalar projection of $\vec{PS}$ onto the unit normal: $d = \left| \vec{PS} \cdot \frac{\mathbf{n}}{|\mathbf{n}|} \right|$
  • How to read: “The distance d is equal to the absolute value of the dot product of the vector PS and the unit normal vector.”
  • Meaning: Shortest distance from point $S$ to the plane—scalar projection onto the normal.

Connected Concepts

• Cylinders in Space
• Arc Length in Space
• Lines in Space