Unmanned aerial vehicles (UAVs) have a wide range of applications in both civilian and military sectors. The UAV guidance system is important for achieving the proposed mission. Traditional Unmanned Aerial Vehicle (UAV) guidance laws often rely on look-ahead points, limiting their ability to track complex 3D paths accurately. This paper proposes a novel look-ahead vector guidance law that depends on differential geometry to enhance path-tracking performance, by arc length representation of the curves. the look-ahead vector encodes both direction and curvature information, enabling smoother and more precise manoeuvres, improving tracking accuracy in tight turns, and simplifying control law calculations. There are two design parameters of the proposed guidance law. Numerical simulations are performed on the 3_D kinematic model of a UAV with two input states by linking the output of the proposed guidance law and the angles of the model. The proposed algorithm was evaluated using MATLAB Simulink simulation for a UAV model. The results demonstrate the Nonlinear Differential Geometry Path-following Guidance Law (NDGPFG) and design parameters effect of the proposed guidance law. Simulation results presented the performance of the proposed look-ahead vector guidance law, paving the way for improved UAV path tracking in various applications.