Do the orbits of the flow of a vector field change when multiplied by a function?

Question

Assume we have a vector field $X$ on open subset $U\subseteq\mathbb{R}^n$ containing the point $0 \in \mathbb{R}^n$, and let $f: U\rightarrow\mathbb{R}$ be a smooth function. I want to know whether flows of $X$ and $fX$ have the same orbits or not.

By orbit, I mean an orbit of the action of the one-parameter group of diffeomorphisms $\theta^X_t$ or $\theta^{fX}_t$.