Vector-valued function

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A graph of the vector-valued function r(t) = <2cost,4sint,t> indicating a range of solutions and the vector when evaluated near t=19.5

A vector-valued function is a mathematical function that maps real numbers onto vectors. Vector-valued functions can be defined as:

$\mathbf{r}(t)=f(t)\mathbf{{\hat{i}}}+g(t)\mathbf{{\hat{j}}}$ or
$\mathbf{r}(t)=f(t)\mathbf{{\hat{i}}}+g(t)\mathbf{{\hat{j}}}+h(t)\mathbf{{\hat{k}}}$

where f(t), g(t) and h(t) are functions of the parameter t, and î, ĵ, and k̂ are unit vectors. r(t) is a vector which has its tail at the origin and its head at the coordinates evaluated by the function.

The vector shown in the graph to the right is the evaluation of the function near t=19.5 (between 6π and 6.5π; i.e., somewhat more than 3 rotations). The spiral is the path traced by the tip of the vector as t increases from zero through 8π.

Vector functions can also be referred to in a different notation:

$\mathbf{r}(t)=\langle f(t), g(t)\rangle$ or
$\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle$

The comma-delimited items within angle-brackets in the notation above is a representation of a column matrix. This notation implies multiplication by a row matrix which consists of unit vectors:

$\mathbf{r}(t)=f(t)\mathbf{{\hat{i}}}+g(t)\mathbf{{\hat{j}}} = \begin{bmatrix} \mathbf{{\hat{i}}} ~~ \mathbf{{\hat{j}}} \end{bmatrix} \cdot \begin{bmatrix} {f(t)}\\ {g(t)}\\ \end{bmatrix}.$