Showing each component of velocity in a helix

Question

In this animation, I want to show each component of velocity (V_x, V_y and V_z) with arrows in the respective direction that should change dynamically i.e the length of the arrows should represent the magnitude of their vectors.

Animate[ParametricPlot3D[{Cos[t], Sin[t], t/10}, {t, 0, u}, 
   Mesh -> {{u}}, MeshStyle -> Red, 
   Method -> {"BoundaryOffset" -> False}, 
   AxesLabel -> {"X", "Y", "Z"}, 
   PlotTheme -> {"Scientific", "BoldColor"}, 
   PlotRange -> {{-2, 2}, {-2, 2}, {0, 8}}] /. 
  Point -> (Sphere[#, .18] &), {u, 10^-6, 30}, AnimationRate -> 0.05]

Answer 1

• For a velocity vector v={a,b,c}, we use {a,b,c}.IdentityMatrix[3] to get it's components vx={a,0,0},vy= {0,b,0},vz= {0,0,c}.

• The vz={0,0,1/10} looks too small.

Clear[f, v, frame];
f[t_] = {Cos[t], Sin[t], t/10};
v[t_] = f'[t];
frame[t_] = v[t]*IdentityMatrix[3];
Animate[Show[
  ParametricPlot3D[{Cos[t], Sin[t], t/10}, {t, 0, u}, 
   AxesLabel -> {"X", "Y", "Z"}, 
   PlotTheme -> {"Scientific", "BoldColor"}, 
   PlotRange -> {{-2, 2}, {-2, 2}, {0, 8}}], 
  Graphics3D[{Arrowheads[.01], 
    Thread@{{Green, Orange, Red}, 
      Arrow[{f[u], f[u] + #}] & /@ frame[u]}}]], {u, 10^-6, 30}, 
 AnimationRate -> 0.05]

Simplify[Sqrt[v[t] . v[t]], Assumptions -> t ∈ Reals]

Sqrt[101]/10

So the norm of the velocity is a constant.

Answer 2

If I understand you right. You can adjust the scale and the arrow type as needed.

Manipulate[
 Module[{p1, p2, xa, ya, za, scale = 2},
  p1 = ParametricPlot3D[{Cos[t], Sin[t], t/10}, {t, 0, u},
     Mesh -> {{u}}, MeshStyle -> Red,
     Method -> {"BoundaryOffset" -> False},
     AxesLabel -> {"X", "Y", "Z"},
     PlotTheme -> {"Scientific", "BoldColor"},
     PlotRange -> {{-2, 2}, {-2, 2}, {0, 8}}, ImageSize -> 500] /. 
    Point -> (Sphere[#, .1] &);
  xa = {{Red, Arrowheads[Small], 
     Arrow[{{Cos[u], Sin[u], u/10}, {scale*Cos[u], Sin[u], u/10}}]}, 
    Text["x", {scale*Cos[u], Sin[u], u/10}, {1, 0}]};
  ya = {{Red, Arrowheads[Small], 
     Arrow[{{Cos[u], Sin[u], u/10}, {Cos[u], scale*Sin[u], u/10}}]}, 
    Text["y", {Cos[u], scale*Sin[u], u/10}, {1, 0}]};
  za = {{Red, Arrowheads[Small], 
     Arrow[{{Cos[u], Sin[u], u/10}, {Cos[u], Sin[u], scale*u/10}}]}, 
    Text["z", {Cos[u], Sin[u], scale*u/10}, {1, 0}]};
  p2 = Graphics3D[{xa, ya, za}];
  Show[p1, p2]
  ],
 {{u, 1, "u"}, 10.^(-6), 100, .001, Appearance -> "Labeled"},
 TrackedSymbols :> {u}
 
 ]