“Adding” Scalars and Bivectors: Ridiculous?! A video for high-school teachers of Geometric Algebra

High-school teachers will be key players in the spread of Geometric Algebra. Providing these teachers with the background to answer students' questions is an important task for us. Here's a high-school-level answer to an objection/question that intelligent students are likely to raise.

LinkedIn group "Pre-University Geometric Algebra"
The group is a space for the sharing and collaborative development of resources for teaching GA at the high-school level. More-advanced materials are welcome, especially if prepared with the additional intent of helping high-school students develop the abilities needed to understand and do such work themselves within a few years.

Document "About the "Addition" of Scalars and Bivectors in Geometric Algebra"

ABSTRACT
“You can’t add things that are of different types!” This objection to the “addition” of scalars and bivectors—which is voiced by physicists as well as students—has been a barrier to the adoption of Geometric Algebra. We suggest that the source of the objection is not the operation itself, but the expectations raised in critics’ minds by the term “addition”. Indeed, the ways in which this operation interacts with others are unlike those of other “additions”, and might well cause discomfort to the student. This document explores those potential sources of discomfort, and notes that no problems arise from this unusual “addition” because the developers of GA were careful in choosing the objects (e.g. vectors and bivectors) employed in this algebra, and also in defining not only the operations themselves, but their interactions with each other. The document finishes with an example of how this “addition” proves useful.

[davidrysdam]

This got super confusing at 11:45. "In the 2D case we are operating in a single plane. That vector will be perpendicular to the plane." Do you mean parallel?

Then it gets even more confusing. "B is just a chunk of that plane. Then B and v are parallel." How can B and v be parallel if v is perpendicular to the plane that B is a chunk of? At this point I can only conclude that you did mean "parallel".

Then we get to "The product of B and v is just a vector." You say this like it's obvious but the only products you've previous mentioned are inner and outer. An inner product of two vectors would be a scalar, an outer product would be a bivector. I guess the inner product of a bivector and a vector might be a vector...? It certainly isn't obvious.

[chrstfer]

Not a teacher, just a guy who left hs 10+ years ago and im wondering where this was in my HS career. Wouldve made things like differential geometry much easier to grok

[ARBB]

I've never heard any satisfactory answer to what a multivector is or does in intuitive terms, because I think it is very much like the idea of explaining what a tensor is.

You use these sums of subspaces of different exterior algebras and of these objects to represent systems, is what I'd say.

No one thinks of the metric and strain tensors as the same thing, but of different objects within the same "control building blocks", and the same can be said of multivectors.

[joepike]

15:35 It might have been easier to understand if you gave the arbitrary numbers to the scalar u dot v and u wedge v and the vector v to show an example of u being solved.

[denki]

10:21 For the "no closure" problem, Isn't the full space the linear combination of all scalars, vectors, bivectors, etc? So adding a scalar and a bivector together still results in a geometric object.

For an analogy, the set of even numbers are closed under addition, so are threeven (multiples of three) numbers. You can add them and result in a number divisible to neither, just one of them, or both, but the result is still surely an integer, which is closed under addition as well.

The analogy breaks down since the two are "the same" kind of object, just with different special properties, but I hope it gets the point.

[joepike]

The problem is the context of which "like things" get added and that is completely dependent on what is on the other side of the equal sign. If on the other side of the equal sign was a scalar or a bivector, then there would be every reason to complain about adding unlike units on the other side. What must therefore be understood is that in the vector multiplication is to be understood an implied mixed units of the other side of the equal sign that must therefore be matched. In a certain sense it is like when an n-th degree polynomial is solved for each degree of term on the other when one is doing partial fraction decomposition.