In[]:=
CurvesAndTheirTangentVectorsEGExSt221::usage="
​​References: Internet: personal.math.ubc.ca, 1.6 Curves and their tangent vectors, Exercise Stage 2, Prob.- 21; Personal.";
Development (a):
In[]:=
2
2
(tSin[πt])
+2
2
(tCos[πt])
+
2
(
2
t
)
==24;
In[]:=
Reduce[2
2
t
+
4
t
==24,t]
Out[]=
t-2||t2||t-
6
||t
6
In[]:=
r[2]={0,2,4};
Development (b):
In[]:=
r[t_]:={tSin[πt],tCos[πt],
2
t
};
In[]:=
T[2]=
r'[2]
Norm[r'[2]]
Out[]=

2π
17+4
2
π
,
1
17+4
2
π
,
4
17+4
2
π

Development (c):
In[]:=
F[x_,y_,z_]:=2
2
x
+2
2
y
+
2
z
-24;​​g[x_,y_,z_]=Grad[F[x,y,z],{x,y,z}];​​n=g[0,2,4]
Out[]=
{0,8,8}
The vectors {0, 8, 8} y

2π
17+4
2
π
,
1
17+4
2
π
,
4
17+4
2
π

they are not parallel, so the curve Γ and the ellipsoid do not intersect at right angles.
In[]:=
Animate[Show[{Graphics3D[{Blue,Point[r[t1]],Arrowheads[.03],Arrow[Tube[{r[t1],r[t1]+r'[t1]}]]}],​​Graphics3D[{Green,Point[r[2]],Arrowheads[.03],Arrow[Tube[{r[2],r[2]+{0,8,8}}]]}],Graphics3D[{Black,Ball[r[2],0.2]}],Graphics3D[{Black,Ball[r[t1],0.3]}],ContourPlot3D[2
2
x
+2
2
y
+
2
z
==24,{x,-6,6},{y,-6,6},{z,-6,6},MeshFalse,PlotRangeAll,ContourStyleOpacity[.5]],​​ParametricPlot3D[r[t],{t,0,t1},ColorFunction"CherryTones",PlotStyleThick]},​​AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll],{t1,0.1,5,0.1},AnimationRunning->False]
Out[]=
t1