The Tangent Vector and Space Curve

An Example

Background Mathematica Stuff

Needs["VisualLA`"]

Off[ParametricPlot3D::"ppcom"];
Off[General::"spell1"];
Off[Power::"infy"];
Off[∞::"indet"];
Off[ParametricPlot3D::"pplr"];
Off[ParametricPlot3D::"ppx"];
Off[Show::"gcomb"]

The Vectors and Required Functions.

The Position Vector

The Parabola

The Helix

r[t_] = {Cos[t], Sin[t], t}; 
    WINDOW = {{-2, 2}, {-2, 2}, {-1, 4  π + 1}}; 
    tRange = {t, 0, 4  π, 0.2}; xMin = 0;

The Sin(x) Curve.

A Trefoil Knot

Graphing Functions

PosVec[t_] := 
        DrawVector3D[r[t], PlotRange -> WINDOW, Ticks -> False, 
          DisplayFunction -> Identity];
SpaceCurve[t_] := 
        ParametricPlot3D[r[s], {s, xMin - 0.1, t}, 
          DisplayFunction -> Identity];

The Position Vector

Table[Show[{PosVec[t]}, DisplayFunction -> $DisplayFunction], 
        Evaluate[tRange]];

The Position Vector and Space Curve

Table[
        Show[{ParametricPlot3D[r[s], {s, - .01, t}, 
              PlotRange -> WINDOW, Ticks -> False, 
              DisplayFunction -> Identity], PosVec[t]}, 
          DisplayFunction -> $DisplayFunction, 
          ViewPoint -> {2.607, Sin[((-t)/8)],  3.000 + Sin[(t/8)]}, 
          AspectRatio -> 1], Evaluate[tRange]];

The Space Curve

Table[
        Show[{ParametricPlot3D[r[s], {s, - .01, t}, 
              PlotRange -> WINDOW, Ticks -> False, 
              DisplayFunction -> Identity]}, 
          DisplayFunction -> $DisplayFunction, 
          ViewPoint -> {2.607, Sin[((-t)/8)],  3.000 + Sin[(t/8)]}, 
          AspectRatio -> 1], Evaluate[tRange]];

Projections onto the coordinate planes

Show[
        GraphicsArray[{Show[{ParametricPlot3D[r[s], {s, - .01, 4  π}, 
                  PlotRange -> WINDOW, Ticks -> False, 
                  DisplayFunction -> Identity]}, 
              ViewPoint -> {0,  π,  0}, AspectRatio -> 1, 
              PlotLabel -> "xz plane"], 
            Show[{ParametricPlot3D[r[s], {s, - .01, 4  π}, 
                  PlotRange -> WINDOW, Ticks -> False, 
                  DisplayFunction -> Identity]}, 
              ViewPoint -> {-π, 0,  0}, AspectRatio -> 1, 
              PlotLabel -> "yz plane"], 
            Show[{ParametricPlot3D[r[s], {s, - .01, 4  π}, 
                  PlotRange -> WINDOW, Ticks -> False, 
                  DisplayFunction -> Identity]}, 
              ViewPoint -> {0, 0,  π}, AspectRatio -> 1, 
              PlotLabel -> "xy plane"]}], 
        DisplayFunction -> $DisplayFunction];