So in my physics coursebook there are two different kinds of derivation of $\frac{dv}{dt}$ of a particle rotating in a circle. Most of you will know these, they are what is called centripetal/radial acceleration $\frac{v^2}{R}$ and tangential acceleration $R\alpha$. Suppose $R$ is radius, $l$ is arc-length, $\theta$ is angular position in radians, and $\omega$ is angular velocity, then the first derivation is something like $$a_{rad}=\frac{dv}{dt}=\frac{v\,d\theta}{dt}=\frac{v\,dl}{R\,dt}=\frac{v^2}{R}\,(=R\omega^2),$$ and the second $$a_{tan}=\frac{dv}{dt}=R\frac{d\omega}{dt}=R\alpha.$$ Now that's clear and all until you start asking questions about how starting from the same origin and unpacking the same derivation actually gives you two very different results.
So I was trying to figure this out, and I thought that maybe it would clear up if I tried doing this with vector quantities, as then you can assume in the tangent case that the unit velocity-vector $\vec v_u$ is constant, and in the radial case that speed $v$ is constant, and variable the other way around. Then $\vec a_{tan}$ is easy: $$\vec a_{tan}=\frac{dv}{dt}\vec v_u=R\frac{d\omega}{dt}\vec v_u=R\alpha\,\vec v_u,$$ but I get stuck with the radial case: $$\vec a_{rad}=\frac{d\vec v}{dt}=\frac{v\,d\vec v}{dt}=R\omega\frac{\,d\vec v}{dt}=\ldots$$$$\vec a_{rad}=\frac{d\vec v}{dt}=\frac{v\,d\vec v_u}{dt}=R\omega\frac{\,d\vec v_u}{dt}=\ldots$$ That's about where I'm at. I've looked for other answers but didn't find the one.