Plane

Definition

Let $n \geq 3$ be an integer. A plane in the Euclidean space $\mathbb{R}^n$ is a set of the form $$P = \{\, p + s u + t v \;:\; s, t \in \mathbb{R} \,\},$$ where $p$ is a point of $\mathbb{R}^n$. Here $u$ and $v$ are linearly independent vectors of $\mathbb{R}^n$.

Remarks

Three points that are not on one line lie in exactly one plane. In $\mathbb{R}^3$, the planes are also the solution sets of the linear equations $a x + b y + c z = d$ with $(a, b, c) \neq (0, 0, 0)$. See the equation of a plane. In the language of affine geometry, a plane is an affine subspace of dimension 2. The word plane also names the whole Euclidean plane $\mathbb{R}^2$.

Explanations

L1 human

A plane in space is like a sheet of glass that goes on forever. To describe it, pick one point on it and two directions in it that are not parallel. Each point of the plane is the start point plus some amount of the first direction plus some amount of the second.