We investigated the F + HD(v = 1, j = 0) → HF + D reaction using the crossed molecular beam technique combined with the D atom Rydberg tagging time-of-flight spectroscopy. By detecting the products at various scattering angles for different collision energies in the range of 0.8-1.2 kcal/mol, we observed the forward-scattering products of HF(v' = 4) and determined the threshold energy for the opening of this reaction channel. Similar experiments were conducted for the F + HD(v = 0, j = 0) → HF + D reaction within the range of 1.1-1.6 kcal/mol, where forward-scattering products of HF(v' = 3) were observed, and the threshold energy for this reaction channel was determined as well. Furthermore, we measured the differential cross-sections for the F + HD → HF + D reaction in both the vibrational ground state and the excited state of HD and analyzed the vibrational quantum-state distribution of the HF products. It was found that the population of vibrational quantum states of the HF products increases synchronously with the excitation of the reactant HD vibrationally.