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## Multivariable calculus

### Course: Multivariable calculus > Unit 2

Lesson 10: Curl- 2d curl intuition
- Visual curl
- 2d curl formula
- 2d curl example
- Finding curl in 2D
- 2d curl nuance
- Describing rotation in 3d with a vector
- 3d curl intuition, part 1
- 3d curl intuition, part 2
- 3d curl formula, part 1
- 3d curl formula, part 2
- 3d curl computation example
- Finding curl in 3D
- Symbols practice: The gradient

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# Describing rotation in 3d with a vector

Learn how a three-dimensional vector can be used to describe three-dimensional rotation. This is important for understanding three-dimensional curl. Created by Grant Sanderson.

## Want to join the conversation?

- It feels like that this is not 3-D rotation, but 2-D rotation only. If you align the axis of rotation in z-direction, and then consider individual points, they are just moving in x or y direction. Very similar to something moving around a point in the x-y plane. If I am correct in visualizing this, can someone help me with an example where a point moves in all 3 directions, or is that not possible at all? As, when I think of it now, if I just consider one single point rotating around ANY axis, you can always align that axis to z-direction and then point will be moving in x-y plane only. I am little confused, please help(6 votes)
- Any
**momentary**rotation is 2D in the sense that it must have an axis of rotation and everything rotates inside a plane that's perpendicular to that axis.

I think your confusion comes from thinking about "crazier" rotations but if you think about it all they really do is changing that axis of rotation with time, and that can also be described by allowing the curl to change and depend on t.(2 votes)

- At3:40, What does Grant mean by putting vectors anywhere in space? Wont the rotation always take place around the axis of rotation (which is why the right hand rule works) which is essentially the vector arrow will be pointing?(1 vote)
- He's talking about the location of the arrow that represents the vector, as a whole... not about the tip of the arrow. You can place the arrow at the origin, or at the poles of the sphere, or anywhere, as long as the arrow has the same direction and magnitude, it is the same vector.(2 votes)

- When I think about a sphere rotating (like a tennis ball or something), I think about how it could be rotating about a certain axis, but then that axis itself could be rotation. Like maybe in 3-space it's rotating around the vector (1, 0, 0) in one instant, then around (0, 1, 0) the next, then around (-1, 0, 0), then around (0, -1, 0), then back to (1, 0, 0), etc. Will 3d curl describe this situation too?(1 vote)
- Yes, this kind of rotation composition will effectively be a rotation too. It is possible to decompose a single rotation, in multiple rotations too. For example, every rotation can be decomposed as a series of rotations around each axis, for reference, this is called Euler Angles (https://en.wikipedia.org/wiki/Euler_angles).(1 vote)

- This doesn't appear to help me sadly, as wouldn't you too have to define the angle the axis it's rotating around is facing? Though the reason this isn't helping may be because I didn't come to learn about describing something with a moving rotation, I only wanted to learn how to describe an X of 45, a Y of 45, and a Z of 45 as just an angle and a magnitude.(1 vote)
- Vectors in rectangular coordinate form is as common as those in polar coordinate form as you require. The transform is easy, like the magnitude of (45,45,45) is equal to sqrt(45^2+45^2+45^2)=45sqrt(3), and its angle to, say, the xOy plane is arctan(1/sqrt(2)). I believe it's somewhere in Precalculus.(1 vote)

- Have to say, whoever this instructor is, he is wayyyyyy better than Sal.(0 votes)

## Video transcript

- [Voiceover] How do you describe rotation in three dimensions? So for example, I have here a globe and it's rotating in some way and there's a certain
direction that it's rotating and a speed with which it's rotating. And the question is how could
you give me some numerical information that perfectly
describes that rotation? So you give me some numbers,
and I can tell you the speed and the direction and
everything associated with this rotation. But before talking about that, let's remind ourselves
of how we talked about two dimensional rotation. So I have here a little pi creature, and I set him to start rotating about and the way that we can describe this, we pretty much need to
just give a rate to it. And you might give that rate
as a number of rotations per second, some unit of
time. So rotations per second. And in this case, I think
I programmed him so that he's going to do one rotation
for every five seconds. So his rotational rate would be 0.2. But that's a little bit ambiguous because if you just say,
"Hey, this little pi creature is rotating at 0.2 rotations per second," someone could say, "Well, is it clockwise or counterclockwise?" So there's some ambiguity. And the convention that
people have adopted is to say, "Well, if I
give you a positive number, if the number is positive,
then that's going to tell you that the nature of the
rotation is counterclockwise, but if I give you a negative number, if instead you see something
that's a negative number of rotations per second,
that would be rotation the other way, going clockwise." And that's the convention. That's just what people have decided on. And with this it's very nice
because given a single number, just one number, and it could
be positive or negative, you can perfectly describe
two dimensional rotation. And there's a minor nuance here, usually in physics and
math, we don't actually use rotations per unit second but
instead you describe things in terms of the number of
radians per unit second. And just as a quick
reminder of what that means, if you imagine some kind of circle, and it could be any circle,
the size doesn't really matter, and if you draw the radius to
that and then ask the question how far along the circumference
would I have to go such that the arc length,
that sort of sub-portion of the circumference, is
exactly as long as the radius? So if this was R, you'd want
to know how far you have to go before that arc length is also R. And then that, that angle,
that amount of turning that you can do, determines one radian. And because there's exactly two pi radians for every rotation, to
convert between rotations per unit second and
radians per unit second, you just multiply this guy by 2π so it would be whatever the
number you have there times 2π. And the specific numbers
aren't too important. The main upshot here is
that with a single number, positive or negative, you
can perfectly describe two dimensional rotation. But if we look over here at the 3D case, there's actually more information than just one number that
we're going to need to know. First of all, you want to know the axis around which it's rotating,
so the line that you can draw such that all rotation
happens around that line. And then you want to
describe the actual rate at which it's going. You
know, is it slow rotation or is it fast? So you need to know a direction
along with a magnitude. And you might say to yourself,
"Hey, direction? Magnitude? Sounds like we could use a vector." And in fact, that's what we do. We use some kind of vector whose length is going to correspond to the
rate at which it's rotating, typically in radians per second, it's called the angular velocity. And then the direction describes
the axis of rotation itself But similar to how in
two dimensions there was an ambiguity between clockwise
and counterclockwise, if this was the only convention we had, it would be ambiguous whether
you should use this vector or if you should use one pointing
in the opposite direction. And the way I've chosen to
draw these guys, by the way, it doesn't matter where they are, remember a vector it just has
a magnitude and a direction and you can put it anywhere in space. I figured it was natural enough to just kind of put them around the poles just so that you could see them on the axis of rotation itself. So the question is,
what vector do you use? Do you use the one
pointing in this direction? Or do you use this green one pointing in the opposite direction? And for this, we have a convention known as the right-hand rule. So I'll go ahead and
bring in a picture here to illustrate the right-hand rule. What you imagine doing
is taking the fingers of your right hand and curling them around in the direction of rotation. And what I mean by that is
the tips of your fingers will be pointing the direction that the surface of the sphere would move. Then when you stick out your thumb, that's the direction that
is the choice of vector which should describe that rotation. So in the specific example we have here, when you stick out your right thumb, that corresponds to the white
vector, not the green one. But if you did things
the other way around, whoops, move this a little
bit. Get him to stay in place. If you move things the other way around, such that the rotation were going kind of in the opposite direction, then when you imagine curling
the fingers of your right hand around that direction, your
thumb is going to point according to the green vector. But with the original rotation
that I started illustrating, it's the white vector, the white vector is the one to go with. And this is actually pretty cool, right? Because you're packing
a lot of information into that vector. It tells you what the axis is. It tells you the speed of
rotation via its magnitude. And then the choice of which
direction along the axis tells you whether the
globe is going one way or if it's going the other. So with just three numbers, the three dimensional
coordinates of this vector, you can perfectly describe any one given three dimensional rotation. And the reason I'm talking
about this, by the way, in a series of videos about curl, is because what I'm about to talk about is three dimensional curl
which relates to fluid flow in three dimensions and
how that induces a rotation at every single point in space. And what's going to happen
is you're going to associate a vector with every single point in space to answer the question
what rotation at that point is induced by the certain fluid flow? And I'm getting a little
bit ahead of myself here. For right now you just need to focus on a single point of rotation and a single vector corresponding to that. But it's important to kind
of get your head around how exactly we represent
this rotation with a vector before moving on to the notably
more cognitively intensive subject of three dimensional curl. So with that, I will see you next video.