Question

If vectors could be curved, what would be possible?

Answer 1

we could describe circular motion with a single vector

Answer 2

You'd have to introduce some clear definitions first. What do you mean by "curved"? and "vector"? How would arithmetic work? (if it's even a part of this ... what would you call it, new branch of mathematics?)

Answer 3

Well I purposefully left it open ended @SithsAndGiggles since I was just curious what people thought. Sure we'd have to introduce clear definitions. Is the curvature of the vector variable? Could we perhaps consider this curvature a parameter so that we can describe two dimensions with 1 vector by scaling the length and bend of it? Or perhaps like this: with a spiral where you have a set spiral that you can increase or decrease how tight the spiral is to get to points inbetween the base spiral. You'd have multiple points that map to the same point, but this isn't that different from how in polar coordinates when r=1 and theta=0 we can get to the same point of r=-1 and theta=pi. We just need to limit it appropriately so we don't have too much overlap.

This might be interesting and or useful in solving differential equations, where the phase field can take on saddle, spirals, or other shapes and there are already theorems that say there are linearly independent solutions on these paths so making linearly independent basis vectors for these paths is already fairly solid.