vg = {vgx[t],  vgy[t],  vgz[t]}

r = {x[t],  y[t],  z[t]}

D[r,  t] == vg

TableForm[
      m1 = {{Cos[θ[t]], -Sin[θ[t]], 0}, {Sin[θ[t]], 
            Cos[θ[t]], 0}, {0, 0, 1}}]

TableForm[
      m2 = {{Cos[ψ[t]],  0,  Sin[ψ[t]]},  {0,  1,  
            0},  {-Sin[ψ[t]],  0,  
            Cos[ψ[t]]}}]

TableForm[m3 = m1 . m2]

TableForm[
      ReplaceAll[
        m3,  {θ[t] -> 0,  ψ[t] -> 0}]]