Show that a curve $r=r(t)$ of class $C^m ; (mgeq 2)$, where $t$ is arbitrary, is a straight line if $r’(t)$ and $r”(t)$ are linearly dependent for all $t$.
So if $r’$ and $r”$ are linearly dependent, that means that there exists a real number $a$ so that $r’(t)=acdot r”(t)$.
If $r$ is parametrised by arc length, we get that $r’(s)=r’(s(t))cdot s’(t)$, thus we have
$$r”(s)=r”(s(t))cdot s’(t) + r’(s(t))s”(t)=r”(s)s’(t)+a r”(s)s”(t)$$
From here we have that $r”(s)(1-s’(t)-s”(t))=0$, so by direct integration of $r”(s)=0$ we get that $r(s)=c_1 t+c_2$.
Is my way of proving correct?