Yes, you have that right. The underlying operations are the same. What you are missing is that the ideas from geometric algebra work in any dimension! Quaternions are a “trick” that only works in 3D.
You can write down Maxwell’s equations using the geometric algebra easily, and it will make them better! They will be obviously coordinate independent.
Pretty much anything with a cross product will be better written using geometric algebra.
You can write down Maxwell’s equations using the geometric algebra easily, and it will make them better! They will be obviously coordinate independent.
Pretty much anything with a cross product will be better written using geometric algebra.