41–43 of 43 posts
Earlier quoted context omitted.
In a geometric setting - if you have two vectors, you can position them so that both have one end at the origin. This spans a plane (test it out yourself in 2D or 3D space with two pencils, put the eraser at the origin for each; it's a parallelogram). The area of the plane will depend on the length of the pencils. You can assign an orientation to the plane by imagining a rotor embedded in the plane that spins either…
Hmmm. "The area of the plane will depend on the length of the pencils". Surely the area of the plane is infinite? The area of the _parallelogram_ will depend on the length of the pencils. And I can't see how "you can assign an orientation to the plane" other than by changing the directions of the pencils. Again this description sounds like it refers to the parallelogram, not the plane. And I don't know what a rotor i…
Quarternions aren't weird. With a 2D angle we use 2 numbers (sin and cos) to calculate vector rotations. It turns out that in 3D we use 4 numbers.
There's a lot of hand waving in that phrase, "it turns out". Sure, "it turns out" that 3D uses four numbers. Why? Geometric algebra explains that in a succinct way that also appeals to our intuition about geometry. Start by using bivectors to represent reflections, then take the closure of your bivectors and you get the even-ordered subalgebra. This will have dimension 2^(N-1)... so 2 for 2D, 4 for 3D, and 8 for 4D.…