---
tag: Z0VBZ906M8
kind: definition
name: Line segment
msc: [52A05, 51N10]
defines: VQ4Q2V45RP
uses: [000AYPA9AJ]
sources:
  - url: https://en.wikipedia.org/wiki/Line_segment
---

## Definition

Let $V$ be a real [[vector space|000AYPA9AJ]] and let $x, y \in V$.
The *line segment* (or closed segment) between $x$ and $y$ is the set
$$[x, y] = \set{(1 - t) x + t y : t \in \R,\ 0 \le t \le 1}.$$
The points $x$ and $y$ are its endpoints.

## Remarks

Example: in $\R^2$, the point $(1, 2)$ lies in $[(0, 0), (2, 4)]$, since $(1, 2) = \tfrac12 (0, 0) + \tfrac12 (2, 4)$ with $t = 1/2 \in [0, 1]$.

Non-example: the point $(3, 6)$ lies on the line through $(0, 0)$ and $(2, 4)$, but $(3, 6) = (1 - t)(0,0) + t (2, 4)$ forces $t = 3/2$, which is not in $[0, 1]$.
So $(3, 6) \notin [(0,0), (2,4)]$.

The segment $[x, y]$ is the set of [[convex combinations|21SXYJYC59]] of $x$ and $y$, and $[x, y] = [y, x]$.
If $x = y$, then $[x, y] = \set{x}$.