Describing rotation in 3d with a vector

How do you describe rotation in three dimensions? So, for example, I have here a globe, and it's rotating in some way. There's a certain direction that it's rotating, a speed with which it's rotating, and the question is how could you give me some numerical information that perfectly describes that rotation?

You give me some numbers, and I can tell you the speed and the direction and everything associated with this rotation. Well, before talking about that, let's remind ourselves of how we talk about two-dimensional rotation. So, I have here a little pie creature, and I set him to start rotating about. The way that we can describe this, we pretty much need to just give a rate to it.

You might give that rate as a number of rotations, number of rotations per second, some unit of time. So, rotations per second. 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 pie 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. 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.

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. Radians! Just as a quick reminder of what that means: if you can, if you imagine some kind of circle, and it could be any circle; the size doesn't really matter.

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.

That angle, that amount of turning that you can do, determines one radian. Because there's exactly two pi radians for every rotation, to convert between rotations per unit second and radians per unit second, you would just multiply this guy by two pi. It would be whatever the number you have there times two pi.

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. 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.

You might say to yourself, "Hey, direction magnitude sounds like we could use a vector." In fact, that's what we do! You 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.

The way I've chosen to draw these guys, by the way, it doesn't matter where they are. Remember, a vector 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? 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. What I mean by that is the tips of your fingers will be pointing kind of 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 and moved this a little bit so let's see, get him to stay in place. If you move things the other way around such that the rotation we're going is 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. 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 that 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. 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.

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?" 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.