Arc length Parameterization

Question

If a given vector $\alpha$ is said to have unit speed such that $\Vert \alpha' \Vert$is equal to one. Does that imply that the vector is paramaterized with respect to arc length?

Answer 1

Yes. Assume that $\alpha$ has unit speed. Then $$L_{s_0}^s[\alpha] = \int_{s_0}^s \|\alpha'(t)\|\,{\rm d}t = \int_{s_0}^s{\rm d}t = s-s_0,$$and $\alpha$ is parametrized by arc-length. Conversely, assume that $\alpha$ is parametrized by arc-length, meaning $$\int_{s_0}^s \|\alpha'(t)\|\,{\rm d}t = s-s_0,$$for all $s$. Differentiating with respect to $s$ gives $\|\alpha'(s)\| = 1$, so $\alpha$ has unit speed.