The transport of ballistic electrons emitted and detected by adjacent point contacts in a two-dimensional electron gas (2DEG) in the system GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As was measured at 1.2 K as a function of the emitter current. Hot carriers with a surplus energy up to 15 meV above the Fermi level were generated by the current flow. It is shown that electron-electron scattering is the main limitation for the quasiparticle lifetime. The experimental data for the ballistic electron propagation from emitter to detector are explained without free parameters by a theory developed by Chaplik and by Giuliani and Quinn. In addition, it is shown that crossing ballistic electron beams in a 2DEG interact with one another, if one of the beams contains hot electrons in the zone of interaction. Experiments on the influence of impurities on the mean free path of ballistic electrons should be done with currents as low as 10 nA. Otherwise, the mean free path contains a contribution from electron-electron scattering. Electron-electron interaction of hot carriers is a serious basic limitation for future devices based on the transport of electrons in the mesoscopic transport regime.