## 17Calculus - Basics of Vector Functions

Using Vectors

Applications

### Partial Integrals

Double Integrals - 2Int

Triple Integrals - 3Int

Practice

### Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Calculus Tools

### Articles

On this page, we discuss what vector functions are and how to determine the domain of vector functions. These ideas will probably not be new to you but how to write vector functions and notation are different than you've probably seen before.

### 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.

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 [49mins-6secs]

video by Dr Chris Tisdell

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]

Practice

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

Problem Statement

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

Solution

### 700 video

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

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

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

Problem Statement

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

Solution

### 697 video

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

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

Problem Statement

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

$$(-1,2]$$

Problem Statement

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

Solution

### 2030 video

video by Krista King Math

$$(-1,2]$$

Log in to rate this practice problem and to see it's current rating.

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. See these separate pages for discussion of limits, derivatives and integrals of vector functions.

### vector functions 17calculus youtube playlist

You CAN Ace Calculus

 parametrics basics of vectors

### Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

### Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Precalculus

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

 The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.