Plane

Builds on Euclidean space Euclidean vector

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 $\mathbb{R}^2$ .

Reading path

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Euclidean space object
Euclidean vector concept
Plane definition

Tag	`0H97Q4Z2SF`
URI	`m2://0H97Q4Z2SF`
Kind	definition
Writing	W0 stub
MSC	51N20
Defines	Plane
Uses	Euclidean space, Euclidean vector

Plane

Definition

Remarks

Reading path

Further reading

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