  We present a systematic derivation of the Biot-Savart equation from the
Nonlinear Schr\"odinger equation, in the limit when the curvature radius of
vortex lines and the inter-vortex distance are much greater than the vortex
healing length, or core radius. We derive the Biot-Savart equations in
Hamiltonian form with Hamiltonian expressed in terms of vortex lines, $$ H=
\frac{\kappa^2}{8 \pi}\int_{|{\bf s}-{\bf s}'|>\xi_*} \frac{d{\bf s} \cdot d
{\bf s}'}{|{\bf s}-{\bf s}'|} \,,$$ with cut-off length $\xi_* \approx
0.3416293 /\sqrt{\rho_0},$ where $\rho_0$ is the background condensate density
far from the vortex lines and $\kappa$ is the quantum of circulation.
