Let $\vec v = (a,b,c)$ be a constant vector and let $\vec r = (x,y,z)$ denote the position vector. Consider the vector field $\vec v \times \vec r$. A straightforward calculation (using determinants or a vector identity) shows that $\nabla \times (\vec v \times \vec r) = 2\vec v$. In order to get a better understanding of this result, a sketch of the vector field helps to convince one that indeed its curl should be $\alpha \vec v$ for some positive constant $\alpha$ .
Question: Are there any further insights from physics, geometry, topology, etc. which help to explain why the constant should turn out to be $\alpha$= 2 ?