---
tag: 018GTRS7A4
kind: concept
name: Vector-valued function
aliases: [vector-function]
msc: [26B12]
uses: [0EPAT51DXQ, 0XPR1E85W0]
sources:
  - url: https://en.wikipedia.org/wiki/Vector-valued_function
    locator: lead section
same_as:
  wikidata: Q1375954
  wikipedia: Vector-valued function
---

## Summary

A vector-valued function is a [[function|0EPAT51DXQ]] whose values are vectors, for example $f \colon D \to \mathbb{R}^m$ with $D \subseteq \mathbb{R}^n$ and $m \ge 2$.
Such a function has $m$ component functions $f_1, \dots, f_m \colon D \to \mathbb{R}$, with $f(x) = (f_1(x), \dots, f_m(x))$ for each $x \in D$.
An example is $r(t) = (\cos t, \sin t, t)$ for $t \in \mathbb{R}$, whose values trace a helix in the [[Euclidean space|0XPR1E85W0]] $\mathbb{R}^3$.

## Remarks

Limits, continuity and derivatives of a vector-valued function into $\mathbb{R}^m$ can be computed one component at a time.
When the domain is an interval of $\mathbb{R}$, the function is a [[parametrised curve|0G7WM8DX4P]].

## Explanations

### L1 human

A vector-valued function gives an arrow, or a point, for each input, not a single number.
If $t$ is time, then $r(t)$ can be the position of a moving particle in space.