Central motion excludes the center

Central motion is the motion of a particle under the influence of a central force. I have known for some year that the particle cannot not pass through the origin. Mimicking a proof in my multivariable calculus book, I'll formalize the fact.

Theorem
If $r:(a, b)\to\mathbb{R}^3$ is twice differentiable, $r(t)\times r''(t) = \vec{0}$ for all $t$ and $r(t_0)\times r'(t_0) \not=\vec{0}$ for some $t_0$, then $r(t)$ is never $\vec{0}$.

Proof
$$\frac{d}{dt}(r(t)\times r'(t)) = r'(t)\times r'(t) + r(t)\times r''(t) = \vec{0}$$ Thus $r(t)\times r'(t)=c$ for some $c$. Because $r(t_0)\times r'(t_0) \not=\vec{0}$, $c\not=\vec{0}$. If $r(t_1)$ were $\vec{0}$ for some $t_1$ then $c$ would be $\vec{0}$ so there is no such $t_1$.

I wonder how to generalize this to higher dimensions.