Linear optimal power flow (OPF) formulations are powerful tools applied to a large number of problems in power systems, e.g., economic dispatch, expansion planning, state estimation, congestion management, electricity markets, among others. This paper proposes a novel mixed-integer linear programming formulation for the AC-OPF of three-phase unbalanced distribution networks. The model aims to minimize the total energy production cost while guaranteeing the network’s voltage and current magnitude operational limits. New approximations of the Euclidean norm, which is present in the calculation of nodal voltage and branch current magnitudes, are introduced by applying a linear transformation of weighted norms and a set of intersecting planes. The accuracy, optimality, feasibility, and scalability of the proposed linearizations are compared with common linear approximations in the literature using two unbalanced distribution test systems. The obtained results show that the proposed formulation is computationally more efficient (almost twice) while being as accurate and more conservative than the benchmarked approaches with maximum errors lower than 0.1%. Thus, its potential application in a variety of distribution systems operation and planning optimization problems is endorsed.