f[t_]:={t^2,t}
v[t_]:=f'[t] v[t]
Clear[motion,tang,unittang]
motion=ParametricPlot[f[t],{t,-3,3}];
tang=vectorPlot[Table[f[i],{i,-3,3}],Table[f'[i]+f[i],{i,-3,3}]];
Show[motion,tang,AspectRatio->Automatic];
T[t_]:=f'[t]/Sqrt[f'[t].f'[t]] T[t]
unittang=vectorPlot[Table[f[i],{i,-3,3}],Table[T[i]+f[i],{i,-3,3}]];
Show[motion, unittang,AspectRatio->Automatic];