Motivating the bivector approac
Rather than thinking about rotations using cross products, the authors reccomend using bivectors. In this view, rotational vectors are represented by a small “tile”, oriented to lie in the plane of rotation, whose area is the magnitude. They assert this means you don’t have to remember any right-hand rules (which I don’t think is particularly hard to begin with), and that this is more natural for coordinate reflections (maybe), and that it is better generalisable to higher dimensions (probably).
Bivectors originate in geometric algebra, but do have applications in (advanced) physics. They can be usefully applied to more basic physics as well.
Okay, research shows students find cross-products and right hand rules very difficult, and very often think that the angular momentum vector points in the plane of rotation.
Bivectors don’t have these problems: the bivector is in the plane of rotation, and there’s no right hand rule to remember.
Additionally, angular momenta are simplified because the bivector “wedge” product is easier to remember, and doesn’t require keeping track of a cyclic coordinate order. Apparently the bivector also corresponds in a straightforward way to the magnitude of a cross-product.
Issues with this approach include the fact that bivector addition is a little complicated, especially compared with tip-to-tail vector addition. Additionally, bivector components can’t be represented as a nice column, has to be as a matrix.
Authors cowardly argue that these aren’t problems at introductory-single-plane problems. Boooooring! Another issue is every existing resource teaches the physics using the pseudovectors, so instructors need to create their own instructional material, perhaps rewriting textbook chapters.
I need to learn how to convert between these bivectors and the traditional pseudovectors.
You can’t teach both; then students complain you’re wasting time/repeating yourself, repeating yourself. Apparently a good way to start is using wedge products as the first step in computing cross products. That seems reasonable! Additionally, you have to teach the vector formalism somehow, since students need to be able to engage with the existing literature.
Traditional introductions to bivectors are only in a geometric algebra context, and do not include their physical interpretation.
Angular Momentum and the Wedge Product
Now we’ll do an example problem involving angular momentum. I’ll do this on my piece of paper, since it isn’t too useful to copy all this down… will write down useful bits.
The bivector of the initial angular momentum can be represented as an oriented “tile” whose attitude in space shows the plane of rotation, and which has an associated orientation specifying the manner of rotation. The area of this tile is the magnitude. The shape is unimportant, we can use whatever shape is convenient (that’s cool!). I don’t like that these manners are just drawn on. Sure, you don’t have to use the RHR or care about system handedness, but you have to draw loads of arrows everywhere…
Wait the wedge product is just the parallelegram from the cross product!! Whaa!! I feel ripped off, I already knew about this. The orientation of a bivector can be found either by rotating the two vectors towards each other, tail to tail, or by putting the vectors tail to tip and that traces out the sides in the right order. … this feels like a RHR but… not.
idk, doing their example problems, it all seems to involve as much little pictures and orderings as vectors. Sure, I don’t have to remember “RHR”… but like that’s really very simple. Especially if you don’t bother with the splayed fingers one and just use the right-hand-grip rule.
Wedge products distribute linearly like other multiplication, and then you can use the rules for unit vectors <-> bivectors.
Wedge products can be between two bivectors, and between two vectors. The result is a bivector.
We can write down the unit bivectors by writing the wedge product between the unit vector along their edge. The bivector is the orientation that takes the first unit vector onto the second. Again, this feels not-that-different?
idk, i guess I’m not every going to be in the position of teaching this so maybe I should focus on just enjoying the physics and learning new maidk, i guess I’m not every going to be in the position of teaching this so maybe I should focus on just enjoying the physics and learning new maths.
Okay, normal vectors to tiles, with lengths equal to the areas of the tiles are the corresponding cross product. ## Bivector Matrices Bivectors can also be written as a matrix of components by enumerating the bivector component formulae, which i’m not sure I can write down on my website… time to find out! The components of the wedge product $\ell = \vector{r} \wedge \vector{p}$ then
ℓ_ij = r_ip_j − r_jp_i
Maybe that worked. It’s nifty that the indices are the same for the first component as they are for the component. Anyway, one component is equal to the projection of the bivector onto the plane defined by that bivector. So the projection onto the xy plane is ℓ_xy.
Okay, so that reflection thingy is that once cross products are defined in terms of bivectors, the behaviour under reflection is a lot less arbitrary since the components of the bivectors just change in the right way. It’s getting late now and I want to be finished with this xD
The procedure for adding two bivectors is quite complicated, or you can just add their associated normal pseudovectors (IE, the cross products) like you’re used to. Geometrically, you reshape the bivectors into rectangles with an edge in common, then place them edge to edge like that, and the third face of the triangular prism formed is the sum.
I guess the thing is I don’t find axis of rotation that hard to understand compared to planes of rotation, and I don’t know how many other students do either. Most rotating things I can think of have an axis, physically.
You can do some nifty stuff with “dot”(matrix) products to go from a bivector rotation and vector displacement to a vector velocity.
Relativistic Angular Momentum
In special relativity, you represent positiosn as 4-vectors with a ct component, and momentums as 4 vectors with an energy E/c component. Cross products aren’t defined in 4 dimensions… but you know what is?? That’s right, 4-bivectors.
these are matrices which are the wedge/outer product of the two component 4 vectors, so we can have angular momentum ez pz.
This matrix transforms under the standard way such matrices do in special relativity (not that I know what that is).
The components of relativistic angular momentum depend on the choice of spacetime origin… huh.
I like the ways they suggest instructors could introduce these ideas, by talking more about areas and rotation planes. e.g., by labelling the torque t_xz rather that t_y.
My takeaways
- This is all very interesting and I’m glad I’ve been tricked into learning about bivectors and their physical interpretation!
- I don’t think this is a meaningfulyl useful pedagogical approach…
- I have however been usefully reminded that angular momentum and so on are pseudovectors, and acquire a weird minus sign under reflection. That’s important to remember.
- Bivectors very cool!