Section 3.1 - Intro to Vector-Valued Functions

Section Objectives

1. Recognize and use equations for vector-valued functions.
2. Sketch and/or describe the graph of a vector-valued function.
3. Find limits of vector-valued functions.
4. Determine the points of continuity of a vector-valued function.

Vector-valued functions

A vector-valued function is a function whose range is a set of vectors. For now, we will mostly be interested in vector-valued functions of a single variable. Such functions can be written in the form

where $x$, $y$, and $z$ are functions of $t$.

The domain of a vector-valued function is the largest set of real $t$-values for which each component is defined.

Examples

• Consider the vector-valued function $\vec{r}(t)=t\hat{\imath} +t^2 \hat{\jmath} +t^3 \hat{k}$. Determine the domain of $\vec{r}$, and evaluate $\vec{r}$ at several values of $t$ in its domain.

• Consider the vector-valued function $\displaystyle \vec{r}(t)=\sqrt{t} \, \hat{\imath} +\frac{t}{t+5} \, \hat{\jmath} + e^t \, \hat{k}$. Determine the domain of $\vec{r}$, and evaluate $\vec{r}$ at several values of $t$ in its domain.

Graphs of vector-valued functions

The graph of a vector-valued function is the set of all terminal points of the vector outputs (when the output vectors are in standard position). Another, and perhaps better, way to think about the graph of a vector-valued function is to recognize that a vector-valued function is essentially a set of parametric equations:

The graph of the function is the graph of the parametric equations.

Examples

• Carefully describe the graph of $\vec{r}(t)=(3t+1) \hat{\imath} + (2t-7)\hat{\jmath} + t \, \hat{k}$. (Hint: What are the corresponding parametric equations?)

• Plot some points on the graph of $\vec{r}(t)=2t \, \hat{\imath} + 4t^2 \, \hat{\jmath}$. Do you recognize the graph?

• To eliminate the parameter means to find a single equation for the graph of a set of parametric equations. For example, if we eliminate the parameter for the parametric equations $x=2t$ and $y=4t^2$, we find that $y=x^2$. We often use algebraic or trigonometric techniques to eliminate parameters. Eliminate the parameter to find a rectangular equation for the graph of $\vec{r}(t)= \cos t \, \hat{\imath} + \sin t \, \hat{\jmath}$.

• Describe the graph of $\vec{r}(t) = \cos t \, \hat{\imath} + \sin t \, \hat{\jmath} + t \, \hat{k}$.

• Graphing technology can be very helpful for graphing vector-valued functions (i.e., parametric equations). Use graphing technology to graph the helix defined by $\vec{r}(t) = \cos t \, \hat{\imath} + \sin t \, \hat{\jmath} + t \, \hat{k}$.

var("t")
parametric_plot3d( [cos(t),sin(t),t] , (t,0,4*pi) )

Limits and continuity of vector-valued functions

Without actually giving a formal definition for the limit of a vector-valued function, let's just say we will think about limits component by component. That is, if $\vec{r}(t)=x(t)\hat{\imath} + y(t)\hat{\jmath} + z(t)\hat{k}$, then the limit of the vector-valued function at $t=c$ is

provided each component limit exists.

Using a similar, component-by-component, approach, we will say the a vector-valued function is continuous at $t=c$ if each component function is continuous at $t=c$. More formally,

• Let $\vec{r}(t)=(t^2-3t+4) \, \hat{\imath} + (4t+3) \, \hat{\jmath}$. Calculate $\displaystyle \lim_{t \to 3} \vec{r}(t)$.

• Let $\displaystyle \vec{r}(t)=\frac{2t-2}{t-1} \, \hat{\imath} + \frac{t}{t^2+1} \, \hat{\jmath} + \frac{9t^2+1}{t} \, \hat{k}$. Calculate $\displaystyle \lim_{t \to 1} \vec{r}(t)$. For what value(s) of $t$ does the limit not exist?

• Referring to the function directly above, at which points is $\vec{r}$ discontinuous? Are those discontinuities removable?