---
tag: 0399QFDFS8
kind: concept
name: Parametric equation of a line
aliases: [vector-equation-of-a-line]
msc: [51N20]
sources:
  - url: https://en.wikipedia.org/wiki/Parametric_equation
    locator: Straight line
  - url: https://en.wikipedia.org/wiki/Line_(geometry)
    locator: Parametric equation
---

## Summary

A parametric equation of a [[line|0MP9BCF6DP]] describes its points as the values of a function of one real parameter $t$.
The line in $\mathbb{R}^3$ through the point $(x_0, y_0, z_0)$ with the nonzero direction vector $(a, b, c)$ has the parametric equations
$$x = x_0 + a t, \quad y = y_0 + b t, \quad z = z_0 + c t, \qquad t \in \mathbb{R}.$$
In vector form, the line through the point $p$ with the direction vector $v \neq 0$ is the set of the points $p + t v$ for $t \in \mathbb{R}$. The same form works in $\mathbb{R}^n$ for each $n \geq 2$.

## Remarks

The line through two distinct points $A$ and $B$ has the parametric equation $A + t(B - A)$. The values $0 \leq t \leq 1$ give the segment from $A$ to $B$, and $t = 1/2$ gives its midpoint.
If $a$, $b$ and $c$ are all nonzero, you can remove $t$. The result is the symmetric form $\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}$.

## Explanations

### L1 human

Think of $t$ as time and of $p + t v$ as the position of a particle. At time $0$ the particle is at $p$, and it moves with the constant velocity $v$. Its path is the line.