Plane
GeometryA plane is a flat, two-dimensional surface that extends infinitely in all directions, with no thickness.
Formula
ax + by + cz = d \text{ (equation of a plane in 3D)}
Definition
A plane is a perfectly flat surface that extends forever in all directions. It has length and width but no thickness at all. Think of an endless, perfectly flat tabletop that goes on forever.
Example
A sheet of paper is like a small piece of a plane. The floor of your classroom represents a plane - except a true plane would extend past the walls forever in every direction.
Key Insight
A plane is two-dimensional: it has two directions to move (left-right and forward-backward) but no up-down thickness. This is what separates it from 3D solid objects.
Definition
A plane is a flat, two-dimensional surface extending infinitely in all directions with no thickness. A plane is uniquely determined by three non-collinear points, or by a line and a point not on that line. Planes are named by three points (plane $ABC$) or by a single capital letter (plane $P$).
Example
Points $A(0,0,0)$, $B(1,0,0)$, and $C(0,1,0)$ determine the $xy$-plane in 3D space. Any two intersecting lines lie in exactly one plane. Parallel planes never intersect.
Key Insight
Two distinct planes either are parallel (never meet) or intersect in exactly one line. This intersection line property is key in 3D geometry and is used in architectural and engineering design.
Definition
A plane in $\mathbb{R}^3$ is an affine subspace of dimension $2$, expressed as $\{P + su + tv : s, t \in \mathbb{R}\}$ for a point $P$ and two linearly independent vectors $u, v$. Equivalently, it is the solution set of $ax + by + cz = d$, where $(a, b, c)$ is the normal vector. In projective geometry, a plane is dual to a point.
Example
The plane through points $A, B, C$ has normal $n = (B - A) \times (C - A)$ (cross product). Its equation is $n \cdot (r - A) = 0$. The distance from point $Q$ to the plane is $|n \cdot (Q - A)| / |n|$.
Key Insight
The duality between points and planes in projective geometry - where any theorem about points and lines has a dual theorem about planes and lines - is a deep structural symmetry. In differential geometry, tangent planes extend this concept to curved surfaces.