In my script, I am reading about the case of a small infinitesimal rotation and it's approximation. If $R$ is the geometrical rotation, and we consider a vector $\vec{OM}$,an infinitesimal angle $d\alpha$, and for simplicity a rotation around the z-axis, then (in the script) we have:
$$R_{\vec u}(d\alpha)\vec{OM}=\vec{OM} + d\alpha \vec u\times \vec{OM}$$
But, as I said $\vec u = \vec e_z$
Then:
$$R_{\vec {e_z}}(d\alpha)\vec{OM}=\vec{OM} + d\alpha \vec {e_z}\times \vec{OM}$$
The formula it seems to me as incorrect. If we consider a counterclockwise (pozitive) rotation around the z-axis, and$\vec{OM}= b\vec e_y$. If we perform the infinitesimal rotation, $\vec {OM}$ will change into a vector that has a pozitive x-component. But if you would apply the above formula for this case of a rotation, you'd end up having another vector with a negative x-component. So what is it that I am doing wrong?