Line

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.

Explanations

L1 human

Start at a point $p$ and walk forward or backward in the direction $v$, for any amount of time $t$. The points you can reach form the line. For example, with $p = (0, 1)$ and $v = (1, 2)$ you get the line $y = 2x + 1$ in the plane.