\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \)
\( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \)
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17calculus > vector functions

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Vector Functions

on this page: ► what are vector functions?     ► domain     ► calculus     ► smoothness

On this page, we discuss what vector functions are, how to perform calculus operations on them and the idea of smooth vector functions. None of these ideas will be new to you, just how we write and interpret the notation.

Difference Between Vector Functions, Vector-Valued Functions and Vector Fields

These three terms are easily confused and some books and instructors interchange them. In general, vector functions are parametric equations described as vectors. Vector fields usually define a vector to each point in the plane or in space to describe something like fluid flow, air flow and similar phenomenon. Vector-valued functions may refer to either vector functions or vector fields. Look carefully at the context and check with your instructor to make sure you understand what they are talking about.

In all three cases, you need to look at the context to see what is being discussed. To avoid confusion, we do not use the term vector-valued function on this site but some of the instructors in the videos we use refer to vector-valued functions.

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What Are Vector Functions?

Vector functions are a way of writing parametric equations of a set of points in the plane or space in vector form. So, for example, if we have a set of parametric equations with parameter \(t\)

\( x(t) = \cos(t) \)     \( y(t) = \sin(t) \)

we can write this as the vector function \(\vec{v}(t)\)

\( \vec{v}(t) = x(t)\hat{i} + y(t)\hat{j} \)

or more directly as

\( \vec{v}(t) = \cos(t)\hat{i} + \sin(t)\hat{j} \)

Key - - The key to using this notation is that the terminal point of the vector defines the \((x,y)\) value for each particular \(t\) value. So, if you think about it, this is just a different way to write parametric equations.

Note - - We use the word 'function' here rather loosely since, as you remember from parametrics, the resulting graphs probably will not pass the vertical line test and, therefore, cannot be correctly referred to 'functions'. However, we are following the standard terms used in most textbooks. So we will continue to call these vector functions whether or not the graphs pass the vertical line test.

3-space - - The idea is the same in 3-space. We will just add a \(\hat{k}\) component so that the equation might look like
\(\vec{w}(t) = f(t)\hat{i} + g(t)\hat{j} + h(t)\hat{k} \)

Multiple parameters - - Note that we are not limited to just one parameter. When describing planes, we may have 2 parameters or even more. So we may have a vector function that looks like
\( \vec{A}(\lambda,\mu) = F(\lambda,\mu)\hat{i} + G(\lambda, \mu)\hat{j} + H(\lambda,\mu)\hat{k}\)

The important point to remember about vector functions is that the terminal point of the vector defines the points in the plane or in space and writing the equations as a vector is just convenient and compact notation that you already learned with parametric equations. You can do everything with vector functions that you can with parametric equations.

Okay, so vector functions are not that hard. They are just a matter of taking parametric equations and writing them in vector form. This first video explains this in more detail, showing how to graph vector functions and it contains some great examples. It's a bit long but well worth taking the time to watch to get this clear in your head.

Dr Chris Tisdell - vector functions [49min-6secs]

The position of an object in the plane (2-dim) or in space (3-dim) can be described by vector functions using the same ideas as above. We go into more detail on the projectile motion page but here is a video to watch first to give you a better feel for vector functions and for what is coming up.

Khan Academy - Position vector valued functions [7min-44secs]

Domain of a Vector Function

The domain of a vector function is the intersection of domain of each term. To find the domain of a vector function, find the domain of the \(\hat{i}\), \(\hat{j}\) and, if it exists, \(\hat{k}\) terms and then take the intersection of those domains. Here are some good practice problems.

Practice 1

Find the domain of \(\displaystyle{ \vec{r}(t) = \frac{t-2}{t+2}\hat{i} + \sin(t)\hat{j} + \ln(9-t^2)\hat{k} }\).


Practice 2

Find the domain of \(\vec{r}(t)=\langle\sqrt{4-t^2},e^{-3t},\ln(t+1)\rangle\).



Practice 3

Determine if any of the points \( (3,e^5, \ln(4))\), \((-1,1,0) \), \( (1/3, e^2, \ln(5)) \) lie on the curve \(\displaystyle{ \vec{r}(t) = \frac{1}{t^2-1}\hat{i} + e^t\hat{j} + [\ln(t+1)]\hat{k} }\)


Calculus of Vector Functions

Okay, so now you know what vector functions are and how to graph them (from the video above). Let's discuss calculus on vector functions. The main topics we will discuss are limits, derivatives and integrals. These are all critical topics that you need to understand when you get to vector analysis.

Limits of Vector Functions

Start with this video on limits of vector functions. This video contains great explanations and examples.

Dr Chris Tisdell - limits of vector functions [44min-37secs]

Let's try some practice problems before we go on.
Unless otherwise instructed, evaluate these limits giving your answers in exact form.

Practice 4

\(\displaystyle{\lim_{t \to 0}{ \left\langle \frac{e^t-1}{t}, \frac{\sqrt{1+t}-1}{t}, \frac{3}{1+t} \right\rangle }}\)


Practice 5

Find \(\displaystyle{\lim_{t\to4}{\vec{r}(t)}}\) for \(\vec{r}(t)=(4-t)\hat{i}+\) \( \left(\sqrt{12+t}\right)\hat{j}- \) \([\cos(\pi t/8) ]\hat{k}\)



Practice 6

Find \(\displaystyle{\lim_{t\to\infty}{\vec{r}(t)}}\) for \(\displaystyle{\vec{r}(t)=\frac{\sin(t)}{t}\hat{i}+\frac{t+1}{3t+4}\hat{j}+\frac{\ln(t^2)}{t^3}\hat{k}}\)



Practice 7

\(\displaystyle{\lim_{t\to 0}{\left(e^{-3t}\vhati+\frac{t^2}{\sin^2t}\vhatj+\cos 2t\vhatk\right)}}\)



Practice 8

\(\displaystyle{\lim_{t\to 0}{\left( e^t\vhat{i}+\frac{\sin t}{t}\vhat{j} \right) }}\)



Practice 9

\(\displaystyle{ \lim_{t\to\infty}{ \left( \frac{2}{t}\vhat{i} + \frac{t^3}{2t^3-8}\vhat{j} + e^{-t}\vhat{k} \right) } }\)



Practice 10

\(\displaystyle{ \lim_{t\to\pi/4}{ \left( \sin^2(t)\vhat{i} + \tan(t)\vhat{j} + \frac{1}{t}\vhat{k} \right) } }\)



Derivatives of Vector Functions

Derivatives of vector functions also require special techniques. This video clip shows some good examples and explains derivatives well.

Dr Chris Tisdell - derivatives of vector functions [13min-36secs]

Try your hand at these practice problems.
Unless otherwise instructed, calculate the derivative of these vector functions. If a value is given, calculate the derivative at that value also.

Practice 11

\(\displaystyle{\vec{r}(t)=[\cos(\pi t)]\hat{i}+\left[\frac{e^t}{t^2}\right]\hat{j}+4t^3\hat{k}}\)



Practice 12




Practice 13

\(\vec{r}(t) = \langle t\sin t,t^2,t\cos 2t\rangle\)



Practice 14

\(\displaystyle{\vec{r}(t)=\frac{5}{t^2}\vhat{i}-4\sqrt{t}\vhat{j}}\),   \(t=1\)



Practice 15

\(\vec{r}(t)=3\cos(t)\vhat{i}+2\sin(t)\vhat{j}-t^2\vhat{k}\),   \(t=\pi/2\)



Practice 16

\(\vec{r}(t)=3\tan(7t)\vhat{i}+\sin^2(t)\vhat{j}-4\ln t\vhat{k}\)



Integrals of Vector Functions

Integrals of vector functions also use special techniques. Here is another video clip that should help you a lot.

Dr Chris Tisdell - integrals of vector functions [6min-57secs]

To put this all together, here is a full lecture on derivatives and integrals of vector functions.

Prof Leonard - Calculus 3 Lecture 12.2: Derivatives and Integrals of Vector Functions [2hrs-42mins-18secs]

Before you go on, try these practice problems.
Unless otherwise instructed, evaluate these integrals.

Practice 17

\(\displaystyle{\int_{0}^{\pi/2}{\vec{r}(t)~dt}}\) for \(\vec{r}(t)=[3\sin^2t\cos t]\,\hat{i}+\) \([3\sin t\cos^2t]\,\hat{j}+\) \([2\sin t\cos t]\,\hat{k}\)



Practice 18

\(\displaystyle{ \int{\frac{5}{t^2}\vhat{i} - 4\sqrt{t}\vhat{j}~dt} }\)



Practice 19




Smooth Vector Functions

For many of our calculations with vector functions, we will require that the vector function be smooth. A smooth vector function is one where the derivative is continuous and where the derivative is not equal to zero. This is comparable to what you already know from basic continuity where a graph is continuous and does not contain any sharp corners. Here is a good video clip explaining this in more detail.

MIP4U - Determining Where a Space Curve is Smooth from a Vector Valued Function [1min-35secs]

Okay, so you are ready to work some practice problems on your own.

Practice 20

Determine the values of t where the vector function \(\vec{r}(t)=t^3\vhat{i}-t^5\vhat{j}\) is smooth.



Practice 21

Determine the values of t where the vector function \(\vec{r}(t)=(t^2e^{-t})\vhat{i}-2(t-1)^2\vhat{j}\) is smooth.



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