We simulate vortex-antivortex annihilation in (2+1)-dimensional O(2) scalar field theory, under a variety of initial conditions. In all cases the energy density of Goldstone-boson radiation shows a sin{sup 2}{alpha} dependence on the angle {alpha} from the final approach due to emission from linear acceleration during the approach and from the overlap of the vortex cores. In collisions at a nonzero impact parameter, there is additional radiation due to circular acceleration. We explore the usefulness of an electromagnetic analogy which uses the vortex world lines as dynamic variables. Our calculations are relevant to the annihilation of antiparallel cosmic strings.