Are there any examples of vector functions where the magnitude of the displacement vector is always equal to the magnitude of the velocity vector?

Question

For example, let the displacement vector be:$$r(t)=\begin{bmatrix}f(t)\\g(t)\\h(t)\\\end{bmatrix}$$and let the velocity vector be:$$v(t)=\begin{bmatrix}x(t)\\y(t)\\z(t)\\\end{bmatrix}$$Are there any such vector functions where $\begin{vmatrix}r(t)\\\end{vmatrix}=\begin{vmatrix}v(t)\\\end{vmatrix}$ for all $\{t:t\in R, t>0\}$?

If not, could somebody explain why such functions cannot or do not exist?

I know this might sound like a straightforward question but I have no idea and am genuinely curious.

Answer 1

Consider for instance a circular path , $$r (t)=\begin{bmatrix} \cos t\\ \sin t\\0\end{bmatrix},\ \ \ \ \ t\geq0. $$