Definition Suppose $V$ is a vector space over $\sR$ or $\sC$ , and $L$ is a subset of $V$ . Then $L$ is a line segment if $L$ can be parametrized as $$L = \{ a+tb \mid t\in[0,1]\}$$ for some $a,b$ in $V$ with $b\neq 0$ .

Sometimes one needs to distinguish between open and closed line segments. Then one defines a closed line segment as above, and an open line segment as a subset $L$ that can be parametrized as $$L = \{ a+tb \mid t\in(0,1)\}$$ for some $a,b$ in $V$ with $b\neq 0$ .

If $x$ and $y$ are two vectors in $V$ and $x \ne y$ , then we denote by $[x,y]$ the set connecting $x$ and $y$ . This is , $\{\alpha x + (1-\alpha )y\ | 0 \le \alpha \le 1\}$ . One can easily check that $[x,y]$ is a closed line segment.

Remarks

• An alternative, equivalent, definition is as follows: A (closed) line segment is a convex hull of two distinct points.
• A line segment is connected, non-empty set.
• If $V$ is a topological vector space, then a closed line segment is a closed set in $V$ . However, an open line segment is an open set in $V$ if and only if $V$ is one-dimensional.
• More generally than above, the concept of a line segment can be defined in an ordered geometry.