I see what you are saying.. As a counterpoint, to me, an oriented plane in 3d very much remembers a rotation though! The graphical convention of a plane with a circle arrow is great because it makes it very clear.
To see what a plane does to a vector, you take the product: of course you start with basis vectors and then later show that everything composes linearly.
The very important thing here is that the geometric product should Not be taught as the sum of scalar and cross product. Geometric product is defined by rules on the basis vectors and by linearity!
Now, I admit that when you see that making actual rotations is not as straighforward as one would expect, it’s confusing, and a bit disappointing. But, I think one can believe that bivectors are about rotation somehow, and then go further to the more complex exponential thing.
Idk maybe it’s just personal preferences, i studied quite a bit of math at college, but i never studied quaternions, and now I’m just very glad I didn’t, since this is much better to me!
To see what a plane does to a vector, you take the product: of course you start with basis vectors and then later show that everything composes linearly.
The very important thing here is that the geometric product should Not be taught as the sum of scalar and cross product. Geometric product is defined by rules on the basis vectors and by linearity!
Now, I admit that when you see that making actual rotations is not as straighforward as one would expect, it’s confusing, and a bit disappointing. But, I think one can believe that bivectors are about rotation somehow, and then go further to the more complex exponential thing.
Idk maybe it’s just personal preferences, i studied quite a bit of math at college, but i never studied quaternions, and now I’m just very glad I didn’t, since this is much better to me!