# Plotting a tangent to a parametric space curve(3D) at a certain point

So I have this curve:

r[t_] := {5 Sin[2 t], -t + 4 Cos[3 t], (2 t - 3 Cos[t])}


I am trying to plot a tangent to this curve at t=2. When I try to do the following:

tangent = ParametricPlot3D[r'[t], {t, -5, 5}]


It gives me nothing or just a single point. Am I doing it wrong and if so what's the right way to do it?

• Similar to 18612
– Syed
Feb 17, 2023 at 6:06

A tangent is defined by: start point: r[t] and end point: r[t]+r'[t]. E.g.(the length of the tangent is scaled):

r[t_] = {5 Sin[2 t], -t + 4 Cos[3 t], (2 t - 3 Cos[t])};
bg = ParametricPlot3D[r[t], {t, 0, 2 Pi}];
Manipulate[
Show[Graphics3D[{
Red, Arrow[{r[t], (r[t] + 0.3 r'[t])}]
}, Axes -> True, PlotRange -> {{-9, 6}, {-10, 5}, {-5, 11}}], bg]
, {t, 0, 2 Pi}]


• Simplest and best approach: +1 for specific question. FrenetSerretSystem (overkill by me)…normal and binormal nice bonus but unnecessary for question posed. Feb 17, 2023 at 10:14

You can use FrenetSerretSystem

For example:

r[t_] := {5 Sin[2 t], -t + 4 Cos[3 t], (2 t - 3 Cos[t])}
tg[u_, s_] :=
Arrow[{r[u],
r[u] + s (FrenetSerretSystem[r[t], t][[2, 1]]) /. t -> u}]
Manipulate[
Show[ParametricPlot3D[r[t], {t, 0, 2 Pi}],
Graphics3D[{Red, PointSize[0.02], Point[r[p]], tg[p, 2]}]], {p, 0,
2 Pi}]