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

<<Calculus`Advanced`

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} ]

tangs = Table[Vector[r[s], r[s]+T[s], RGBColor[0,0,1]], {s,-1.2,1.2,0.2}];

norms = Table[Vector[r[s], r[s]+n[s], RGBColor[1,0,0]], {s,-1.2,1.2,0.2}];

binorms = Table[Vector[r[s], r[s]+b[s], RGBColor[0,1,0]], {s,-1.2,1.2,0.2}];

Select the graphics cell below and type command-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,tangs[[i]],norms[[i]],binorms[[i]]], {i,1,13} ]