Defining a vector function in terms of the unit vectors $\bf{i}$, $\bf{j}$, $\bf{k}$

Question

I am given the following equation for the position of a particle:

${\bf r}(t) = \sin(11t)\ {\bf i} + t^4\ {\bf j} + \cos (11 t)\ {\bf k}$

I seek to:

1. Find the equation of the tangent line at $t=0.2$.
2. Graph the tangent line at $t=0.2$, and the path of the particle from $t=0$ to $t=2 \pi/11$ on the same graph. The line segment for the tangent line should be symmetric about the point of tangency.

I'm having trouble even declaring the vector function. I'm trying r[t_] := {Sin[11 t], t^4, Cos[11 t]}, but I keep getting an error message. SetDelayed::write: Tag List in {0.808496,0.0016,-0.588501}[t_] is Protected. >>So my first question would be, how do you properly define a 3D vector function?

After getting a vector function declared, how would I go about solving for $t=0.2$? Would it be as simple as r[0.2] or would I have to use Evaluate[]? I'm very new to the world of programming, as I'm more on the hardware side of the spectrum. Any help would be greatly appreciated.

Answer 1

Your error message is likely due to the fact that you have a prior definition of r or t or both. Hence clear them:

Clear[r,t];

Then:

myposition[t_] := {Sin[11 t], t^4, Cos[11 t]};
Show[
 ParametricPlot3D[myposition[t], {t, 0, 2 π/11}, BoxRatios -> {1,1,1}],
 Graphics3D[{Red, Thickness[0.01], 
     Arrow[{myposition[.2], myposition[.2] + .03 (D[myposition[t], t] /. t -> 0.2)}]}]]

If the line segment (arrow) needs to be symmetric with respect to the point:

myposition[t_] := {Sin[11 t], t^4, Cos[11 t]};
Show[ParametricPlot3D[myposition[t], {t, 0, 2 \[Pi]/11}, 
  BoxRatios -> {1, 1, 1}], 
 Graphics3D[{Red, Thickness[0.005], 
   Arrow[{myposition[.2] - (k = .03 (D[myposition[t], t] /. 
           t -> 0.2)), myposition[.2] + k}]}]]

Or:

Manipulate[
 Show[ParametricPlot3D[myposition[t], {t, 0, 1}],
      Graphics3D[{Red, PointSize[0.02], Point[myposition[t]]}, 
        PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}],
      Graphics3D[{Red, Thickness[0.01], 
         Arrow[{myposition[t], 
         myposition[t] + .03 (D[myposition[tt], tt] /. tt -> t)}]}]],
 {t, 0, 1}]

The tangent line defined at $t = .2$ is:

myline = myposition[.2] + t (D[myposition[tt], tt] /. tt -> 0.2)

(* {0.808496 - 6.47351 t, 0.0016 + 0.032 t, -0.588501 - 8.89346 t} *)