Summary of Vector-Valued Functions and Space Curves

A vector-valued function is a function of the form [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}[/latex] or [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}[/latex] where the component functions [latex]f[/latex], [latex]g[/latex], and [latex]h[/latex] are real-valued functions of the parameter [latex]t[/latex].
The graph of a vector-valued function of the form [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}[/latex] is called a plane curve. The graph of a vector-valued function of the form [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}[/latex] 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
[latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}[/latex] or [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}[/latex], or [latex]{\bf{r}}(t)=\langle{f(t),g(t)}\rangle[/latex] or [latex]{\bf{r}}(t)=\langle{f(t),g(t),h(t)}\rangle[/latex]
Limit of a vector-valued function
[latex]\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}}[/latex] or [latex]\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}}[/latex]

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

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

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

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

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

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

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