How would I find the Binormal vector? [duplicate]

Question

How would I find the Binormal vector if r[t_]:={sin(7t),t^4,cos(7t)} in Mathematica?

This is the Mathematica code I have:

    r[t_] := {Sin[7 t], t^4, Cos[7 t]};
    circle := 
    ParametricPlot3D[r[t], {t, 0, 2 Pi/7}, PlotStyle -> {Thick, 
    Black}]
    utvec[t_] := {r'[t]/sqrt[r'[t].r'[t]]}
    utvec[0.4]
    (r'[0.4])*t + r[0.4]
    Show[circle, 
    ParametricPlot3D[(r'[0.4])*t + r[0.4], {t, 0, 2 Pi/7}, 
    PlotStyle -> {Thick, Blue}]]
    nvec[t_] := {r''[t]/sqrt[r''[t].r''[t]]}
    nvec[0.4]
    (r''[0.4])*t + r[0.4]
    ubnvec[t_] := Cross[utvec[t], nvec[t]]

Also, how can I graph something that looks like this:

Answer 1

We have

r[t_] := {Sin[7 t], t^4, Cos[7 t]}

Now, we can implement directly the definition of the binormal vector

FullSimplify[Cross[r'[t], r''[t]]/Norm[Cross[r'[t], r''[t]]]]

which gives

{(4 t^2 (-7 t Cos[7 t] + 3 Sin[7 t]))/Sqrt[ 2401 + 16 Abs[t^2 (-7 t Cos[7 t] + 3 Sin[7 t])]^2 + 16 Abs[t^2 (3 Cos[7 t] + 7 t Sin[7 t])]^2], 49/Sqrt[ 2401 + 16 Abs[t^2 (-7 t Cos[7 t] + 3 Sin[7 t])]^2 + 16 Abs[t^2 (3 Cos[7 t] + 7 t Sin[7 t])]^2], ( 4 t^2 (3 Cos[7 t] + 7 t Sin[7 t]))/Sqrt[ 2401 + 16 Abs[t^2 (-7 t Cos[7 t] + 3 Sin[7 t])]^2 + 16 Abs[t^2 (3 Cos[7 t] + 7 t Sin[7 t])]^2]}

In case that t is real, we can inform Mathematica about that fact as follows:

FullSimplify[Cross[r'[t], r''[t]]/Norm[Cross[r'[t], r''[t]]], 
 t ∈ Reals]

which results in

{(4 t^2 (-7 t Cos[7 t] + 3 Sin[7 t]))/Sqrt[
 2401 + 144 t^4 + 784 t^6], 49/Sqrt[2401 + 144 t^4 + 784 t^6], (
 4 t^2 (3 Cos[7 t] + 7 t Sin[7 t]))/Sqrt[2401 + 144 t^4 + 784 t^6]}

Answer 2

r[t_] = {Sin[7 t], t^4, Cos[7 t]};
{tangent, normal, binormal} = FrenetSerretSystem[r[t], t][[2]];
t = 1;
Show[ParametricPlot3D[r[t], {t, 0, 2 Pi/5}, PlotStyle -> Yellow], 
 Graphics3D[{AbsoluteThickness[5], White, 
   Arrow[{r[t], r[t] + tangent}], Blue, Arrow[{r[t], r[t] + normal}], 
   Red, Arrow[{r[t], r[t] + binormal}]}], Background -> Cyan, 
 PlotRange -> All, Boxed -> False, Axes -> False, 
 ViewPoint -> {1.20, 2.86, -1.31}, 
 ViewVertical -> {0.25, 0.96, -0.04}]