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. In Mathematics, the real numbers may be described informally in several different ways 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 are unit vectors. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, parametric equations are a method of defining a curve In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length 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. In Linear algebra, a column vector or column matrix is an m × 1 matrix, i This notation implies multiplication by a row matrix which consists of unit vectors:

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

The row matrix is usually omitted (to be inferred by the reader). In Linear algebra, a row vector or row matrix is a 1 × n matrix, that is a matrix consisting of a single row \mathbf In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length The function may thus be written in the following shorthand:

$\mathbf{r}(t)=\begin{bmatrix} {f(t)}\\ {g(t)}\\\end{bmatrix}.$

## Properties

The domain of a vector-valued function is the intersection of the domain of the functions f, g and h. In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently