Line

Builds on Euclidean space Euclidean vector

Definition

Let $n \geq 2$ be an integer. A line in the Euclidean space $\mathbb{R}^n$ is a set of the form $L = \{\, p + t v \;:\; t \in \mathbb{R} \,\},$ where $p$ is a point of $\mathbb{R}^n$ . Here $v$ is a nonzero vector of $\mathbb{R}^n$ . The vector $v$ is a direction vector of $L$ .

Remarks

For two distinct points $A$ and $B$ , exactly one line contains both: the line through $A$ with the direction vector $\overrightarrow{AB}$ . The point $p$ and the vector $v$ are not unique. The same line has each point of $L$ as a base point, and each nonzero multiple of $v$ as a direction vector. In the plane $\mathbb{R}^2$ , the lines are also the sets $\{(x, y) : a x + b y = c\}$ with real numbers $a$ , $b$ , $c$ and $(a, b) \neq (0, 0)$ . In the language of affine geometry, a line is an affine subspace of dimension 1.

Reading path

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Euclidean space object
Euclidean vector concept
Line definition

Tag	`0BMYPJBJH4`
URI	`m2://0BMYPJBJH4`
Kind	definition
Writing	W1 stated
MSC	51N20
Defines	Line
Uses	Euclidean space, Euclidean vector

Line

Definition

Remarks

Reading path

Further reading

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