f[t_]:={t^2-t,t+5Sin[t]};
curve=ParametricPlot[f[t],{t,-3,3},PlotStyle->{Black,Thickness[.03]}];
d=1;
lines=Graphics[Join[{Thickness[.01],Red},Table[Arrow[{f[k],f[k]+f'[k]*d}],{k,-3,3,d}]]];
Show[curve,lines]

f[t_]:={t^2-t,t+5Sin[t]};
curve=ParametricPlot[f[t],{t,-3,3},PlotStyle->{Black,Thickness[.03]}];
d=.25;
lines=Graphics[Join[{Red,Thickness[.01]},Table[Arrow[{f[k],f[k]+f'[k]*d}],{k,-3,3,d}]]];
Show[curve,lines]

f[t_]:={t^2-t,t+5Sin[t]};
curve=ParametricPlot[f[t],{t,-3,3},PlotStyle->{Black,Thickness[.03]}];
d=.1;
lines=Graphics[Join[{Red,Thickness[.01]},Table[Arrow[{f[k],f[k]+f'[k]*d}],{k,-3,3,d}]]];
Show[curve,lines]

ListPointPlot3D[Table[{1+2t,1+4t,(1+2t)(1+4t)},{t,0,1,0.01}],Filling->0]

ListPointPlot3D[Table[{t^2/2,t^3/3,1 + t^8/18},{t,0,2,0.01}],Filling->0]

Show[
VectorPlot[{1,1},{x,-1.5,1.5},{y,-1.5,1.5}],
ParametricPlot[{Cos[t],Sin[t]},{t,0,2Pi},PlotStyle->{Black,Thickness[.015]}]/.
Line[x_]:>{Arrowheads[{0.,0.1,0.1,0.1,0.1}],Arrow[x]}
]

Show[
VectorPlot[{y,-x},{x,-1.5,1.5},{y,-1.5,1.5}],
ParametricPlot[{Cos[t],Sin[t]},{t,0,2Pi},PlotStyle->{Black,Thickness[.015]}]/.
Line[x_]:>{Arrowheads[{0.,0.1,0.1,0.1,0.1}],Arrow[x]}
]

Show[
VectorPlot3D[{y,x^2,x*z},{x,0,3},{y,0,27},{z,-2,25},
VectorPoints->10,VectorScale->.005],
ParametricPlot3D[{t,3t^2,t^3-2},{t,0,3},PlotStyle->{Black,Thickness[.015]}]/.
Line[x_]:>{Arrowheads[{0.,0.1,0.1,0.1,0.1}],Arrow[x]}
]

Show[
VectorPlot[{2y,x},{x,-1.5,1.5},{y,-1.5,1.5}],
ParametricPlot[{t,t},{t,0,1},PlotStyle->{Black,Thickness[.015]}]/.
Line[x_]:>{Arrowheads[{0.,0.1,0.}],Arrow[x]}
]

Show[
VectorPlot[{2y,x},{x,-1.5,1.5},{y,-1.5,1.5}],
ParametricPlot[{t,0},{t,0,1},PlotStyle->{Black,Thickness[.015]}]/.
Line[x_]:>{Arrowheads[{0.,0.1,0.}],Arrow[x]},
ParametricPlot[{1,t},{t,0,1},PlotStyle->{Black,Thickness[.015]}]/.
Line[x_]:>{Arrowheads[{0.,0.1,0.}],Arrow[x]}
]

Show[
VectorPlot[{2y,x},{x,-1.5,1.5},{y,-1.5,1.5}],
ParametricPlot[{Cos[t]+1,-Sin[t]},{t,Pi,3Pi/2},PlotStyle->{Black,Thickness[.015]}]/.
Line[x_]:>{Arrowheads[{0.,0.1,0.}],Arrow[x]}
]

Show[
VectorPlot[{2y,x},{x,-1.5,1.5},{y,-1.5,3.5}],
ParametricPlot[{t + Sin[Pi t]^8 Cos[t]^5,Cos[2 Pi t]^4+2t},{t,0,1},PlotStyle->{Black,Thickness[.015]}]/.
Line[x_]:>{Arrowheads[{0.,0.1,0.}],Arrow[x]}
]

Show[
VectorPlot[{2y,x},{x,-1.5,1.5},{y,-1.5,3.5}],
ParametricPlot[{t + Sin[Pi t]^8 Cos[t]^5,Cos[2 Pi t]^4+2t},{t,0,1},PlotStyle->{Black,Thickness[.015]}]/.
Line[x_]:>{Arrowheads[{0.,0.1,0.}],Arrow[x]},
ParametricPlot[{t,1+2t},{t,0,1},PlotStyle->{Red,Thickness[.015]}]/.
Line[x_]:>{Arrowheads[{0.,0.1,0.}],Arrow[x]}
]

VectorPlot[{2x y, x y},{x,-3,3},{y,-3,3}]

VectorPlot3D[{y z, x z, x y},{x,-3,3},{y,-3,3},{z,-3,3}]

VectorPlot3D[{y,z,x},{x,-3,3},{y,-3,3},{z,-3,3}]