Geometric Algebra -- Angle Between Two Vectors

[PatrickPowers]

In geometric algebra I have a plane defined by a bivector. Given two unit vectors in this plane, how do I find the angle I need to rotate the first vector so that it becomes equal with the second.

The usual method for computing the angle between two vectors gives an answer between zero and pi. That's no good. I need an answer between zero and 2pi. I'd rather not have several cases depending on quadrants. It seems like there has to be a direct way to do it.

[mr_e_man]

Let's call the vectors u and v, and the (normalized) bivector i. We want a complex number z = cos θ + i sin θ such that u z = v. But this is just z = u⁻¹ v = u v = u∙v + u∧v. Now we have cos θ = u∙v and sin θ = -i(u∧v), and you can use atan2 or something similar to get θ.

[PatrickPowers]

Let's call the vectors u and v, and the (normalized) bivector i. We want a complex number z = cos θ + i sin θ such that u z = v. But this is just z = u⁻¹ v = u v = u∙v + u∧v. Now we have cos θ = u∙v and sin θ = -i(u∧v), and you can use atan2 or something similar to get θ.

Oh that's delightful. This was driving me to exhaustion. I looked around the Internet and the closest to a resource I could find was a physicist refering to GA as a cult. There seems to be nowhere else I can get answers to such questions.

My next question is, how did you learn this so I can do the same? I don't want to have to run for help every time I encounter something like this. I have about five GA books but simple things like this aren't discussed. What they do is give me bunch of operators. I have no idea what to do with them, so I learn zero.

[mr_e_man]

I learned some from the websites of Hestenes (section IV of the primer is relevant to your first question), Macdonald, Denker, and maybe others; from wikipedia, and math.stackexchange; and from my own pencil work.

[PatrickPowers]

I learned some from the websites of Hestenes (section IV of the primer is relevant to your first question), Macdonald, Denker, and maybe others; from wikipedia, and math.stackexchange; and from my own pencil work.

That primer is the stuff, terse and down to Earth. I have all of Hestenes' books I could find on line but missed that one.

[PatrickPowers]

Let's call the vectors u and v, and the (normalized) bivector i. We want a complex number z = cos θ + i sin θ such that u z = v. But this is just z = u⁻¹ v = u v = u∙v + u∧v. Now we have cos θ = u∙v and sin θ = -i(u∧v), and you can use atan2 or something similar to get θ.

I learned more from these four sentences than from some entire GA books.