Hey! Just getting into linear algebra and Calc 3, and we're looking allot at unit vectors etc, but the vectors i,j,k etc or w (cross product of u and v) generally still use 1 as their base unit and scale linearly. Last week we got into linear transformations, and it got me thinking.... If a unit vector represented a scaling nonlinear formula that would do all kinds of crazy things! The most obvious example I could think of that reminded me of this was I imagine if the unit vector represented something like a 1/inf at zero and scaled out, you'd almost have something representing spacetime fabric near black holes.

What other way though are nonlinear unit vectors used in real life applications though? It seems like such an interesting concept.