For $\frac{d\hat{\phi}}{dt}$, how is it perpendicular when you take the limit $\Delta t \rightarrow 0$?

I made a geometric argument: the direction difference vector $\vec{\Delta \hat{\phi}}$ is along the line connecting the tips of the two unit $\hat{\phi}$ vectors at two times. The closer the two $\hat{\phi}$'s are to parallel, the closer the line connecting the tips is to being perpendicular to each of them.