Main content

### Course: Multivariable calculus > Unit 1

Lesson 4: Visualizing vector-valued functions# 3d vector fields, introduction

Vector fields can also be three-dimensional, though this can be a bit trickier to visualize. Created by Grant Sanderson.

## Want to join the conversation?

- I'm sorry if I seem ignorant for asking this here, but if you can, may you please show me where to find the 3-d visualization program that you use?(9 votes)
- It is an application that comes with Mac OS X called
*Grapher*. If you are on a Mac, you should be able to find it in the`/Applications/Utilities/`

directory.(19 votes)

- so now it matters where we start drawing the vector, whereas earlier we could move it around?(6 votes)
- That's right, because in this case the place where you start the vector communicates the "input", while the vector itself is the "output". As you say, when you are thinking of a vector off on its own, not as the output of some vector valued function, you can place it wherever you want.

Also, when vector fields come up in physics, there's usually a physical meaning to the location of a vector. For example, it might tell you the velocity of a fluid at that point, so putting it on that point matters.(21 votes)

- I know you can graph these graphs via Grapher (a Mac app), but is there any standalone software (preferably for windows) that can be used to graph these complex functions?(3 votes)
- I think you might be able to do it with MatLab!(1 vote)

- i am tripping on the last example.

if output is the identity, then why isn't it a positional vector?i am having hard time wrapping my head around why it is from the input point.(2 votes)- It's not really an identity. An identity usually means 1, unless identity is used in a different way in math somewhere I am not aware of. Which is totally possible.

I am not sure if there is a word for it, but the idea of this one is you draw a vector to any point, and the vector field at that point will be the vector from the origin to there, but now with the starting point at the point (x,y,z) So it almost doubles the length, or in other words makes a vector <2x, 2y, 2z>, though the first half of those vectors would be erased.

Let me know if that doesn't make sense.(1 vote)

- When he shows f(x, y, z) = [1, 0, 0], why are there so many of the same vector starting at different points? Is it saying for any x, y, z (point) we have [1, 0, 0] coming out from that point?(2 votes)
- Just realised, that compared to the previous video of 2-dimensional space vector field (Vector Field Introduction at2:16).. this one maps to itself. A point f(2, 2, 2) would map to (2, 2, 2) and there would be no particles flowing outwards, isn't it?

If, otherwise, the output space is supposed to be added to the input space (as it appears to be in the video), the current representation does not indicate that it's a vector field transformation.(2 votes) - So when we are plotting f(1, 1, 1 ) = [1, 1, 1], we take (1, 1, 1) as starting point, add [1, 1, 1] from it and connect them. Am I guessing correct?(2 votes)
- So vector fields could be used to create an effect of gravital attraction of a planet in a videogame? Like using the player position respect to a certain planet to determine the velocity vector.(2 votes)
- I like to think of it more as simulating the flow of a body of fluid.(1 vote)

- How do you find 3D graph stimulators on iPad?(1 vote)
- GeoGebra grapher should be satisfying for you

https://itunes.apple.com/us/app/geogebra-math-calculators/id687678494(2 votes)

- So which program were you using?(1 vote)

## Video transcript

- [Voiceover] So in the last video, I talked about vector
fields in the context of two dimensions, and here,
I'd like to do the same but for three-dimensions. So a three-dimensional vector field is given by a function, a
certain multi-variable function that has a three-dimensional input given with coordinates x, y and z, and then a three-dimensional vector output that has expressions that
are somehow dependent on x, y, and z, I'll just
put dots in here for now, but we'll fill this in with
an example in just a moment. And the way that this works, just like with the
two-dimensinal vector field, you're gonna choose a
sample of various points in three-dimensional space. And for each one of those points, you consider what the
output of the function is and that's gonna be some
three-dimensional vector. And you draw that vector
off of the point itself. So to start off, let's
take a very simple example, one where the vector that outputs is actually just a constant. So in this case, I'll make
that constant the vector, one, zero, zero. So what this vector is,
it's just got a unit lenth in the x direction, so this is the x axis. So all of the vectors
are gonna end up looking something like this where it's a vector that has length one in the x direction. And when we do this, at
every possible point, well not every possible point, but a sample of a whole bunch of points, we get a vector field
that looks like this. At any given point in space, we get one of these little blue vectors and all of them are the same, they're just copies of each other, each pointing with unit
length in the x direction. So as vector fields go,
this is relatively boring, but we can make it a
little bit more exciting if we make the input
start to depend somehow on the actual input. So what I'll do to start,
I'll just make the input y, zero, zero. So they're still just gonna
point in the x direction, but now it's gonna depend on the y value. So let's think of a second
before I change the image, what that's gonna mean. The y axis is this one here, so now the z axis is pointing
straight in our face, that's the y. So as y increases value
to one, two, three, the length of these
vectors are gonna increase, it's gonna be a stronger
vector in the x direction, a very strong vector in the x direction. And if y is negative, these
vectors are gonna point in the opposite direction. So let's see what that looks like. Here we go. So in this vector field, color and length are used to indicate the
magnitude of the vector. So red vectors are very long, blue vectors are pretty short, and at zero, we don't even see any because those are
vectors with zero length. And just like with two
dimensional vector fields, when you draw them, you lie a little bit. This one should have a
length of one, right? Because when y is equal to one, this should have a unit length, but it's made really, really small. And this one up here,
where y is five or six, should be a really long vector, but we're lying a little bit because if we actually drew them to scale, it would really clutter up the image. So a couple things to
notice about this one, since the output doesn't depend on x or z, if you move in the x direction, which is back and forth here, the vectors don't change. And if you move in the z direction which is up and down, the
vectors also don't change. They only change as you
move in the y direction. Okay, so we're starting to
get a feel for how the output can depend on the input. Now let's do something
a little bit different. Let's say that all three of the components of the input depend on x, y, and z, but I'm just gonna make it
kind of an identity function. At a given point x, y, z, you output the vector itself, x, y, z. So let's think about what
this would actually mean. And let's say you've got a given point, some point floating off in space. What is the output vector for that? Well the point has a certain x component, a certain y component, and a z component. And the vector that corresponds to x, y, z is gonna be the one from the
origin to that point itself. Let me just draw that here from the origin to the point itself. And because of how we do vector fields, you move this so that instead
of stemming from the origin, it actually stems from the point. But the main thing to take away from here is it's gonna point directly
away from the origin. And the farther away the point is, the longer this vector will be. So with that, let's take a look
at the vector field itself. Here we go. So again, you kind of
lie when you draw these. Like the vectors, these red
guys that are out at the end, they should be really long 'cause this vector should
be as long as that point is away from the origin. But to give a cleaner vector field, you scale things down,
and notice the blue ones that are close to the center here, are actually really, really short guys. And all of these are pointing
directly away from the origin. And this is one of those vector fields that is actually pretty, a good one to have a strong intuition of 'cause it comes up now and then, thinking about what the
identity function looks like as a vector field itself. In the next video, I'll
talk through another example that's a little bit more
complicated than this and can hopefully give
an even stronger feel for how the output can
depend on x, y, and z.