Animation of the Unit Tangent, Normal, and Binormal
Vectors Moving Along the Twisted Cubic Curve

Norm[v_List]  :=  Sqrt[v . v]

Vector[a_List, b_List, c___] := 
      Graphics3D[{Thickness[0.01], c, Line[{a, b}]}]

r[t_] = {t,t^2,t^3};

T[t_] = r'[t]/Norm[r'[t]] //Simplify

n[t_] = T'[t]/Norm[T'[t]] //Simplify

B[t_] = Cross[T[t],n[t]] //Simplify

curve = ParametricPlot3D[ {t,t^2,t^3}, {t,-1.2,1.2},
	PlotRange->{{-2,2},{-1,2.5},{-3,3}},
	ViewPoint->{3,0.5,1.5} ]

tangents = Table[Vector[r[t], r[t]+T[t], RGBColor[0,0,1]], {t,-1.2,1.2,0.1}];

normals = Table[Vector[r[t], r[t]+n[t], RGBColor[1,0,0]], {t,-1.2,1.2,0.1}];

binormals = Table[Vector[r[t], r[t]+B[t], RGBColor[0,1,0]], {t,-1.2,1.2,0.1}];

Select the graphics cell below and type Alt-y  to animate it. The blue vector isthe unit tangent, the red vector is the unit normal vector, and the  green vector is the unit binormal vector.

Do[Show[curve, tangents[[i]], normals[[i]], 
        binormals[[i]]], {i, 1, 25}]