Summary of Vector-Valued Functions and Space Curves

A vector-valued function is a function of the form ${\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}$ or ${\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}$ where the component functions $f$ , $g$ , and $h$ are real-valued functions of the parameter $t$ .
The graph of a vector-valued function of the form ${\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}$ is called a plane curve. The graph of a vector-valued function of the form ${\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}$ is called a space curve.
It is possible to represent an arbitrary plane curve by a vector-valued function.
To calculate the limit of a vector-valued function, calculate the limits of the component functions separately.

Key Equations

Vector-valued function
${\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}$ or ${\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}$ , or ${\bf{r}}(t)=\langle{f(t),g(t)}\rangle$ or ${\bf{r}}(t)=\langle{f(t),g(t),h(t)}\rangle$
Limit of a vector-valued function
$\underset{t\to{a}}{\lim}{\bf{r}}(t)=\left[\underset{t\to{a}}{\lim}f(t)\right]{\bf{i}}+\left[\underset{t\to{a}}{\lim}g(t)\right]{\bf{j}}$ or $\underset{t\to{a}}{\lim}{\bf{r}}(t)=\left[\underset{t\to{a}}{\lim}f(t)\right]{\bf{i}}+\left[\underset{t\to{a}}{\lim}g(t)\right]{\bf{j}}+\left[\underset{t\to{a}}{\lim}h(t)\right]{\bf{k}}$

component functions: the component functions of the vector-valued function ${\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}$ are $f(t)$ and $g(t)$ , and the component functions of the vector-valued function ${\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}$ are $f(t)$ , $g(t)$ and $h(t)$

limit of a vector-valued function: a vector-valued function ${\bf{r}}(t)$ has a limit ${\bf{L}}$ as $t$ approaches $a$ if $\underset{t\to{a}}{\lim}|{\bf{r}}(t)-{\bf{L}}|=0$

plane curve: the set of ordered pairs $\left(f(t),g(t)\right)$ together with their defining parametric equations $x=f(t)$ and $y=g(t)$

reparameterization: an alternative parameterization of a given vector-valued function

space curve: the set of ordered triples $\left(f(t),g(t),h(t)\right)$ together with their defining parametric equations $x=f(t)$ , $y=g(t)$ and $z=h(t)$

vector parameterization: any representation of a plane or space curve using a vector-valued function

vector-valued function: a function of the form ${\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}$ or ${\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}$ , where the component functions $f$ , $g$ , and $h$ are real-valued functions of the parameter $t$