Summary of quantities associated to parametrized motion in (ℝ^3)
D. McClendon

This graphic shows the relative positions of velocity, acceleration and principal unit normal vectors to a parametrized curve in (ℝ^3):

f[t_] = {2.5Sin[t],  1.25Cos[t], t^3/10};
T[t_] = f'[t] /Sqrt[f'[t][[1]]^2 + f'[t][[2]]^2 + f'[t][[3]]^2];
s =2.Pi/3;
NormalVec = T'[s]/Norm[T'[s]];
Show[ParametricPlot3D[{ f'[s]{3t,3t,3t} + f[s], f[3t]},
{t, -1,1}, AxesLabel -> {"x","y","z"}, PlotStyle -> {{Thick,Red, Dashed}, {Thick, Black}}, PlotRange -> {{-3,3},{-3,3},{-3/2,3}}, Boxed -> False, Axes -> False],
Graphics3D[{GrayLevel[.65],Line[{{0,0,0},{0,2,0}}]}],
Graphics3D[{GrayLevel[.65],Line[{{0,0,0},{3,0,0}}]}],
Graphics3D[{GrayLevel[.65],Line[{{0,0,0},{0,0,1.5}}]}],
Graphics3D[{Red, Arrow[{f[s], f[s] + f'[s]}]}],
Graphics3D[{Red, Thickness[.01],Arrow[{f[s], f[s] +T[s]}]}],
Graphics3D[{Black,Sphere[f[s], .075]}],
Graphics3D[{Orange,Thickness[.01],Arrow[{f[s], f[s] +  NormalVec}]}],
Graphics3D[{Orange, Arrow[{f[s], f[s] +3NormalVec}]}],
Graphics3D[{Darker[Green], Arrow[{f[s], f[s] +f''[s]}]}],
Graphics3D[{Darker[Green], Dashing[{.005,.005}],Line[{f[s] + .825f'[s], f[s] + .825f'[s] + 1.875  NormalVec,f[s] +1.875NormalVec}]}],
Graphics3D[{GrayLevel[.65],Text["x",{3,.15,0}]}],
Graphics3D[{GrayLevel[.65],Text["y",{0,2,.15}]}],
Graphics3D[{GrayLevel[.65],Text["z",{.25,.25,1.5}]}],
Graphics3D[{Red, Text[Style["v = f'", Bold], f[s] + f'[s] + {0,-.5,0}]}],
Graphics3D[{Red, Text[Style["T", Bold], f[s] + .2f'[s] + {0,0,.55}]}],
Graphics3D[{Darker[Green], Text[Style["a = f''", Bold], f[s] + f''[s] + {0,.25,.25}]}],
Graphics3D[{Darker[Green], Text["(Subscript[a, N])", f[s] +.9f'[s] +1.15 NormalVec]}],
Graphics3D[{Darker[Green], Text["(Subscript[a, T])", f[s] + .3f'[s] + 2.1 NormalVec]}],
Graphics3D[{Black, Text[Style["x(t)",Bold],  f[s] + {.75,0,0}]}],
Graphics3D[{Black, Text[Style["γ",Bold], f[s+1] + {.5,0,0}]}],
Graphics3D[{Orange, Text[Style["T'",Bold],f[s] + 3.65 NormalVec + {0,-.5,0}]}],
Graphics3D[{Orange, Text[Style["N",Bold],f[s] +  NormalVec + {0,0,-.25}]}]
]