Normal and Tangent of Acceleration in 3D

Question

I am trying to find the normal and tangent of acceleration. I know the formula for the tangent of acceleration is $((Acceleration . Velocity)/(Velocity.Velocity))*Velocity$ and the normal of acceleration is the same but with $Velocity$ replaced with the perpendicular of velocity. I think my errors arise because I am trying to manipulate parametric functions incorrectly.

Anyway, here is my attempt:

Clear[t,x,y,z,P,velocity,acceleration,atan,tanvector];
x[t_] = 2 Sin[t]; 
y[t_] = 6 Sin[t/2]^2;
z[t_] = 3 Cos[t];
P[t_] = {x[t], y[t], z[t]}; 

curveplot = ParametricPlot3D[P[t],{t,1,6},PlotStyle->Thickness[0.01]];
velocity[t_] = {x'[t], y'[t], Z'[t]};
acceleration[t_] = {x''[t], y''[t], z''[t]};

atan[t_] = ((acceleration[t].velocity[t])/(velocity[t].velocity[t]))*velocity[t];
tanvector[t_]:= Vector[atan[t],Tail->P[t],VectorColor->Blue]

Show[curveplot,Table[tanvector[t],{t,1,6,(6-1)/10}],
  PlotRange -> All, AxesLabel -> {"x","y","z"}]

This question is different from this question: Finding unit tangent normal and binormal vectors because I am asking for the tangent and normal of acceleration using the equation: $((Acceleration . Velocity)/(Velocity.Velocity))*Velocity$.

I am missing an undertanding of how to manipulate parametric plots.

Answer 1

You had a typo in the definition of velocity: capital Z instead of z. With this typo fixed, you may be able to get the result you want. However, I can't use your plot commands directly to verify this because Vector isn't defined. So I use a simple replacement by Arrow for this purpose:

Clear[t, x, y, z, P, velocity, acceleration, atan, tanvector];
x[t_] = 2 Sin[t];
y[t_] = 6 Sin[t/2]^2;
z[t_] = 3 Cos[t];
P[t_] = {x[t], y[t], z[t]};

curveplot = 
  ParametricPlot3D[P[t], {t, 1, 6}, PlotStyle -> Thickness[0.01]];
velocity[t_] = {x'[t], y'[t], z'[t]};
acceleration[t_] = {x''[t], y''[t], z''[t]};


atan[t_] = ((acceleration[t].velocity[t])/(velocity[t].velocity[t]))*
   velocity[t];
tanvector[t_] := {Blue, Arrow[{#, # + atan[t]} &@P[t]]}

Show[curveplot, Graphics3D@Table[tanvector[t], {t, 1, 6, (6 - 1)/10}],
  PlotRange -> All, AxesLabel -> {"x", "y", "z"}]