We connected dislocation-based atomic-scale and continuum models of plasticity in crystalline solids through numerical simulations of dislocation intersections in face-centered cubic crystals. The results contradict the traditional assumption that strain hardening is governed by the formation of sessile junctions between dislocations. The interaction between two dislocations with collinear Burgers vectors gliding in intersecting slip planes was found to be by far the strongest of all reactions. Its properties were investigated and discussed using a multiscale approach.