Unit tangent vector

The direction $\mathbf{r}'(t)$ points, with speed divided out: $\mathbf{T}(t)=\mathbf{r}'(t)/|\mathbf{r}'(t)|$ .

The unit tangent vector $\mathbf{T}(t) = \dfrac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}$ splits off just the direction $\mathbf{r}'(t)$ carries, with its "how fast" divided away — a compass reading with the speedometer subtracted out.

Examples

For $\mathbf{r}(t) = \langle 3\cos t, 3\sin t, 4t\rangle$ , $\mathbf{T}(t) = \left\langle -\tfrac35\sin t,\ \tfrac35\cos t,\ \tfrac45\right\rangle$ .