Least-squares projection of filament deformation velocity (${\int }_0^L(g_t+V_1+{\Omega }_{1}x)\left(\genfrac{}{}{0ex}{}{1}{x}\right)dx=0$)

At first order, the transverse drag depends on $g_t+V_1+{\Omega }_{1}x$. Its zero force and torque conditions make this residual orthogonal to $1$ and $x$. Thus $-V_1-{\Omega }_{1}x$ is the linear least-squares projection of the prescribed transverse deformation velocity onto the span of $1$ and $x$. This recasts a force-balance calculation as a projection.