The geometric product is super useful and important! (As well as much nicer to work with when doing algebraic manipulation.)
In particular, it is what lets you take products, inverses and quotients (assuming the denominator is non-null) of arbitrary vectors.
> surprisingly hard to compute
Hm? No it isn’t....
Several times in the last few years I have done several pages of complicated calculations in terms of coordinates or matrices, where the intermediate steps were basically an indecipherable mess of symbols, and then sat down, really thought about what I was doing for a while, and re-written it using GA language and ended up simplifying my work down to a few lines of simple algebraic manipulations, where I could typically then give a nice geometrical interpretation to each step.
If you want to compute the geometric product using concrete numbers in terms of a basis, then it is straightforward to write the code (or work through by hand).
What is hard is not the geometric product. What is hard is geometry! There is a lot of depth and subtlety, and it takes years to really learn your way around. That is not the fault of the language though; there is a certain amount of irreducible complexity involved.
Personally I would love to get some people together and try to write a high-school-accessible geometry / vector algebra book using GA language. It would be a few years of hard work, because figuring out the right order for the curriculum, the right types of problems to build intuition, which tools to include or leave out of an introductory book, how to best show the work for a whole collection of worked examples (ideally with some interactive computer diagrams), and so on.
In particular, it is what lets you take products, inverses and quotients (assuming the denominator is non-null) of arbitrary vectors.
> surprisingly hard to compute
Hm? No it isn’t....
Several times in the last few years I have done several pages of complicated calculations in terms of coordinates or matrices, where the intermediate steps were basically an indecipherable mess of symbols, and then sat down, really thought about what I was doing for a while, and re-written it using GA language and ended up simplifying my work down to a few lines of simple algebraic manipulations, where I could typically then give a nice geometrical interpretation to each step.
If you want to compute the geometric product using concrete numbers in terms of a basis, then it is straightforward to write the code (or work through by hand).
What is hard is not the geometric product. What is hard is geometry! There is a lot of depth and subtlety, and it takes years to really learn your way around. That is not the fault of the language though; there is a certain amount of irreducible complexity involved.
Personally I would love to get some people together and try to write a high-school-accessible geometry / vector algebra book using GA language. It would be a few years of hard work, because figuring out the right order for the curriculum, the right types of problems to build intuition, which tools to include or leave out of an introductory book, how to best show the work for a whole collection of worked examples (ideally with some interactive computer diagrams), and so on.