[adornKey]

Interesting article.

I have been telling people for more than a decade now to use the exterior product (and e.g. get rid of the clutter of strange Minus-Signs and div, grad and rot in Electrodynamics).

And I was really happy to see that people finally start doing that.

But when I saw the Geometric Product, it didn't look like anything I want. If someone says that it looks like a thing that the cat brought in, I'll think about it and will probably agree.

[hamish_todd]

The geometric product does transform composition. It has all the properties you want for multiplication in an algebra, or indeed a monoid or group.

So it is like matrix multiplication, but for transforms represented as multivectors. Multivectors are nicer than matrices because they are made out of the separate (exterior algebra) objects so you can geometrically interpret them. For example, a rotation-reflection (rotoreflection/improper rotation) will have a grade 1 part and a grade 3 part. One of them is the plane you reflect in, one is the point you rotate around.