Arc length and position vector

Question

Suppose $r(t) = (x(t),y(t),z(t))$ be the position vector of a point $(x,y,z)$ is $dr/dt$ and $ds/dt$ the same? Where $s$ is the arc length.

Answer 1

No. $r(t)$ is a vector and $s(t)$ is a scalar. Their derivatives have the same number of components as the functions, so they cannot be the same. You do have $\frac {ds}{dt}=\left|\frac {dr}{dt}\right|$