# Geometric Algebra Resources

Geometric Algebra brings together algebra, geometry, vectors, complex numbers, and linear algebra. It provides a single unification of all elementary math and serves as an excellent basis for physics, robotics and computer graphics. Are there any Geometric Algebra resources available for teaching high school students?

Are there any Vector Algebra resources that include the outer product (wedge product) before the cross product?

• Is this for yourself or for teaching other high school students? If you need more details on how to teach it to others, I would be happy to collaborate with you to create a minimal resource. If it is for yourself, I could suggest some other resources. – Richard Mar 24 '16 at 10:11
• I think 2D wedge products are great to teach for simplifying trigonometry - but are you looking for material on 3D wedge products? Would you be looking to show that wedge products of vectors are of a higher grade rather than simply a perpendicular vector? What are you primarily wanting to demonstrate by using wedge rather than cross products. – Richard Mar 25 '16 at 1:33
• The wedge product is 2D while the cross product is 3D. So the wedge product should be a simpler concept to introduce. And, it is a 2D object that works in any n-dimensional space. The cross product only works in a 3D space. – Allyn Shell Mar 28 '16 at 15:43

I have taught basic Geometric Algebra to high school students. I have not been able to find any appropriate resources. I would only do this with very able students, as otherwise you are risking some serious confusion which could interfere with understanding more basic concepts. As prerequisites, I would suggest that complex numbers and 2D vectors (addition/subtraction) should be understood. I did not need dot product at the most basic level, but I'll need to introduce it when doing trig with GA.

The most difficult concept for students to accept was non-commutative operations. Don't underestimate how big this change is for students. You may be able to demonstrate this with a non-commutative group (the symmetries of a 2D shape including rotations and flips), but this is a rather big detour. You can show how rotating a rubiks cube is non-commutative. I emphasised that they are not allowed to take any shortcuts in rearranging the order of multiplied vectors - just follow the rules:

$$\hat x \hat x=1$$ $$\hat y \hat y=1$$ $$\hat x \hat y= -\hat y \hat x$$

I revised how vectors can be written in terms of basis vectors: $(3,4) = 3\hat x +4\hat y$, and how addition/subtraction works.

Did some simple vector multiplication to demonstrate that distributive and associative laws still work fine, and to practice the anti-commutative property of basis vectors.

I then gave them 3 example vectors: $(1,0),(3,4),(0,1)$ and discussed where they would rotate to. They then had to experiment to find what multiplication could rotate any vector $90^\circ$ anti-clockwise, ie.$(1,0)\cdot \Box = (0,1)$. This was a too hard given the small amount of practice that they had, so I gave them some hints. A demonstration of how $\hat x \hat y$ works can be given (ie. the first $\hat x$ cancels the $\hat x$ of the vector, and then multiply by $\hat y$ to get the desired result), then show it automatically works for the others.

The final step is to ask "what is $(\hat x \hat y)^2$?". After 30 seconds a triumphant student exclaimed "It's $i$!!" (referring to $\hat x \hat y$). Which was of course the whole point of the exercise, to demonstrate why $i$ works - because of the underlying unity of geometric algebra. I did not discuss how complex numbers are confusingly used to encode both rotations and vectors.

I have not introduced dot (symmetric) products or areas and wedge (anti-symmetric) products, though I will touch on this when we get to Trig II. You can easily calculate the cos and sine of the angle between vectors. It is quite beautiful when studying sine and cosine rules - especially the sine rule becomes obvious.

Not a curriculum I'm afraid, but I hope it helps.

• in response to your comment on the Vector Algebra Text page where you said, "For most students, attempting to teach this could be extremely counter productive as it would interfere with their later education when it is someone else's turn to teach them a traditional curriculum." How can introducing a simpler method be confusing or interfere with a student's education? Doing that does not prevent explaining the more difficult cross product. It just makes it easier to explain. – Allyn Shell Mar 25 '16 at 15:15
• P.S. Richard, I do want to thank you for your comments above they really will be very helpful. – Allyn Shell Mar 25 '16 at 15:27
• @AllynShell Sometimes what seems simple for me is not simple for students who are simply trying to learn enough to get through the course. For the more advanced students, I regard GA as an investment in their mathematics understanding which will pay off in the future. For the other students who are quite happy using standard formulae, I'm not sure that they would thank me when the formulae and methodology they have to use in later courses is quite different from what I've taught them. However it is possible that I'm simply being too conservative. Good luck in your endeavours! – Richard Mar 26 '16 at 0:45
• But, it appears that at the college level the engineering and science classes are moving to GA which would put the students ahead. That is why I want to find ways to present this in HS. – Allyn Shell Mar 28 '16 at 15:50
• @AllynShell That is really great news! I have not seen any undergrad courses with Geometric Algebra at universities around me, so either they are behind the curve, or the universities in your area are in front of it. If you could let me know which university you are looking at, I would love to see how they are teaching it, what textbook etc. Also, where I live the HS curriculum does not include 3D vectors or cross product, so maybe we are behind the curve compared to where you are. That is why I'm only introducing 2D wedge product to prove the sin rule. – Richard Mar 29 '16 at 1:29