Section 13.1: Vector Functions and Space Curves

In general, a function is a rule that assigns to each element in the domain an element in the range. A vector-valued function, or vector function, is simply a function whose domain is a set of real numbers and whose range is a set of vectors. We are most interested in vector functions r whose values are three-dimensional vectors. This means that for every number \(t\) in the domain of r there is a unique vector in \(V_3\) denoted by \(\mathbf{r}(t)\). If \(f(t)\), \(g(t)\), and \(h(t)\) are the components of the vector \(\mathbf{r}(t)\), then \(f\), \(g\), and \(h\) are real-valued functions called the component functions of r and we can write \[ \mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle = f(t)\mathbf{i} + g(t)\mathbf{j} + h(t)\mathbf{k} \] We use the letter \(t\) to denote the independent variable because it represents time in most applications of vector functions.

EXAMPLE 1 If \[ \mathbf{r}(t) = \langle t^3, \ln(3-t), \sqrt{t} \rangle \] then the component functions are \[ f(t) = t^3 \qquad g(t) = \ln(3-t) \qquad h(t) = \sqrt{t} \] By our usual convention, the domain of r consists of all values of \(t\) for which the expression for \(\mathbf{r}(t)\) is defined. The expressions \(t^3\), \(\ln(3-t)\), and \(\sqrt{t}\) are all defined when \(3-t > 0\) and \(t \ge 0\). Therefore the domain of r is the interval \([0, 3)\).