Vector-valued function

A function assigning a vector to every input $t$ : $\mathbf{r}(t)=\langle f(t),g(t),h(t)\rangle$ .

A vector-valued function assigns a vector to every real number $t$ in its domain: $\mathbf{r}(t) = \langle f(t), g(t), h(t)\rangle = f(t)\,\mathbf{i} + g(t)\,\mathbf{j} + h(t)\,\mathbf{k}$ . The three ordinary functions $f$ , $g$ , $h$ are $\mathbf{r}$ 's component functions, and as $t$ varies, the tip of $\mathbf{r}(t)$ — always drawn from the origin — sweeps out a curve in space called its graph, or trace. The domain of $\mathbf{r}(t)$ is the intersection of the domains of $f$ , $g$ , and $h$ .

Examples

$\mathbf{r}(t) = \langle 2\cos t, 2\sin t, t\rangle$ traces a helix of radius $2$ climbing the cylinder $x^2+y^2=4$ .
A flight recorder logging position at every time $t$ is a vector-valued function, unlike a bare photograph of the track.