Section 1.1.5: How Vectors Transform
STANDARD DISCLAIMER: I am not a physicist. I do jokes about physics and then people vastly overestimate my level of knowledge. Years ago, I was studying to get a BS in physics, but dropped out because I had to run a comics business. I’m now teaching myself again. As I am not an expert, I may say dumb things. When I do, please correct me. This blog is mostly to make me understand the topics I am studying by pretending to be teaching them. There are lots of people who are better at it. You have been warned.
I’m starting with this section because everything earlier is vector stuff you should know from an intro physics course. This was the first section I remember giving me a little trouble.
The author brings up a more rigorous definition of a vector, which is basically this: a vector is something that can handle component changes.
Here’s how I like to think about it. A vector is a Real Thing. The philosophically-inclined may say I’ve opened up a bigger topic than this blog post can handle. But, screw them. Let me just say this – one quality of a Real Thing should be that it doesn’t change when I change my perspective on it.
Consider an example. Suppose there are stars in space, Star A and Star B. Somewhere between them, far enough from them that gravity doesn’t matter, there is a ball. Suppose the ball is moving from A to B at speed S. These are facts about the system which have nothing to do with whether or not some guy named Descartes came along and invented analytic geometry. Regardless of what coordinate system we throw on the ball, it will remain true that the ball moves along the line from A to B at speed S.
We can change the components of the ball’s velocity by changing the coordinate system. For instance, we could (and probably would) define the path of the ball as going along an axis, thus reducing all other coordinates to 0. We could use some obnoxious made up coordinate system that produced all sorts of ugly vector components. We can do whatever we want in our mathematical brains. BUT, the ball itself will not stop moving from A to B. This movement is a Real Thing.
And, you’re lucky it’s so. If not, every time a physics student reconsidered the coordinates of something, it might change course.
Now, this may seem obvious, but it turns out there are other types of vectory things that don’t behave this way. The example I’m familiar with is rotation vectors, called pseudovectors. If a thing with mass rotates, you can say it has angular momentum. The pseudovector for angular momentum (L) points according to the right hand rule. You might say that this seems liked a Real Thing, in that you can move around the wheel and L always seems to point the same way. Fair enough. But, consider a mirror transformation. In that case, the pseudovector points the wrong way. To give an example:
Suppose you have a pasty on your right nipple. This pasty has a tassle on it, and you’re spinning the tassel so that if there were an arrow on the tassle, it’d point to your face, then your right shoulder, then your armpit, and so on. Now, let’s suppose Alice is in front of your nipple and Bob is behind it. According to the right hand rule, the vector points out from your nipple toward Alice and away from Bob.
Now, imagine we take the left-for-right mirror of this setup. That is, imagine your left nipple perfectly mirrors the behavior of the right nipple, and Alice is still in front of you (flipped) and Bob is still behind you (flipped). You’re doing some pretty impressive tassel-slinging, as the tassel on lefty is rotating opposite to that on righty. If you’re having trouble visualizing, imagine them rotating in the way two touching gears rotate.
Since the rotation is flipped, you use the right hand rule differently. Now, when you do the right hand rule, L is pointing at Bob. So, you changed what L was doing just by changing your coordinate system.
Compare this to our ball moving from Star A to Star B. If we mirror left for right (visualizing it in 2D helps), it’ll look different, but the ball will always be moving from A to B.
Okay, good. I’m always proud when my blog post is actually longer than the section in the book. It’s a sign of virtuoso verbosity.
Now, to the math at hand. Of course, it’s not enough to just SAY the vector is still a Real Thing. If we change the coordinates, we have to have a way to automatically specify what the ball is doing in the new system. In the abstract, the thing you gotta do is modify each component so that you’re still talking about the same old Real Thing.
So, first the book defines the components of a vector A as A_y=Acos(θ) and A_z=Asin(θ).
Simple enough. The y and z components of A are given by decomposing it in the usual way.
Then, we transform coordinates. In this case, we’re rotating counterclockwise a little. Having done so, we have to tell ourselves where the tip of A is, with respect to the two axes. Well, the cute thing is, since we’re just rotating, thanks to the magic of trig, the components are easy to find just by figuring out what the new angle is.
Referencing the diagram, you can see that the new angle between the new y axis and A is smaller than the old angle by ϕ. Thus, the y component in the new system is Acos(θ-ϕ). All we did there was say “in the new system, the rotational distance from the y axis to the tip of A got smaller, and thus the straight line distance (given by the cosine) got smaller.”
So, from Acos(θ-ϕ), we use one of our old trig laws to expand to
Acos(θ)cos(ϕ) + Asin(θ)sin(ϕ)
And, remember a second ago when we said A_y = Acos(θ) and A_z = Asin(θ)? Well, now we can use that simplify down. So, we can say say that in the new system, you can get the new y component of A with this:
A_y*cosϕ + A_z*sin(ϕ)
Neat, right? We can do a similar trick to get the new component of of z.
You might say “Christ! Why do we gotta do all this just to get the new vector? Why’s it look so complicated?!” Well, remember we’re just getting the general case here. And, in the general case for 2D, if you rotate the axes, you gotta make some trades so the Real Thing vector stays the same. If you change your system so the y component is smaller, you gotta borrow from z to make up for it, and vice versa. That’s what A_y*cosϕ + A_z*sin(ϕ) is doing. It’s saying “for your new vector component, take this much of the old y and this much of the old z.” And this makes sense. Not, for example, if ϕ=0 (i.e. there was zero rotation), your new y component is the same as the old, and you don’t have to borrow anything from z. If ϕ=π, you’ve rotated 180 degrees. So, you gotta flip the sign for the y component, but you still don’t borrow anything from z. Make sense?
The book goes on to explain that you can make it matrixy, but there’s no way I’m doing the LaTeX for that here. Having made it matrixy, you can see that that you can do a similar trick in 3D or 4D or whatever. You may note that the number of moves to be made squares. That is, transformation in 2D requires 4 things, transformation in 3D requires 9 things, and so on. Why should this be?
Well, remember, each modifier (e.g. R_xx, R_zy, etc.) is saying “to get the new thing, change me by this.” So, in the case of 2D, you’re starting with two components. Each component needs to be modified in each dimension by some amount. So, that’s 2×2. Similarly, in 3D, you have three components, each of which need to be modified in each dimension. Hence, 3×3 = 9.
At the end Griffiths brings up tensors, though I’m not entirely sure why. It’s fitting with the conversational tone, but I kinda wish there were some more meat on the bone here. Here’s my quick and dirty understanding.
A scalar is a magnitude no direction (e.g. temperature, mass, etc.)
A vector is a magnitude one direction (e.g. velocity, force, etc.)
You can imagine continuing down this path. Consider a third thing which is a magnitude with two directions, or three, or four. All these things are said to be tensors. A tensor with 0 direction is a scalar. A tensor with one direction is the familiar vector. Beyond that, it’s tensor rank ____. Tensor rank 2 has two directions, rank 3 has three directions, and so on.
Again, not sure why it’s coming up here, BUT hopefully there’ll be an obvious answer for me later.
NEXT STOP: Interesting Problems in 1.1.5