TangentUnitaryVector::usage="Find the tangent unitary vector of the curver[t] = 6i + Cos[3t]j + 3 Sin[4t]k, 0 ≤ t < 2π.Reference: Personal. ";
In[]:=
Manipulater[t_]:={6,Cos[3t],3Sin[4t]};v[t_]:=r'[t];​​T1[t_]:=
v[t]
v[t].v[t]
;​​p=Grid[{{Point},{r[t1-0.01]}},FrameAll,FrameStyle->Red,ItemStyle->15,ItemSize15];​​w=Grid[{{TangentUnitaryVector},{T1[t1-0.01]}},FrameAll,FrameStyle->Green,ItemStyle->15,ItemSize15];​​Show[{ParametricPlot3D[r[t],{t,0,t1},PlotStyleDirective[Thick,Black],PerformanceGoal->"Quality"],Graphics3D[{Green,Point[r[t]],Arrowheads[.05],Arrow[Tube[{r[t],r[t]+3T1[t]}]]}]/.tt1-0.01,Graphics3D[{Red,Ball[r[t1],0.1]}]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,PlotRange6,BoxRatiosAutomatic,PerformanceGoal"Quality",ViewPoint{5,-2.4,2.},ImageSize700],Style["Tangent unitary vector:",Bold,Large],{{t1,1,"Value (​
t
1
​)"},0.01,2π,0.1},Delimiter,{{p,3,"Point coordinates r[t]"}},Delimiter,{{w,4,"Tangent unitary vector T[t]"}},ControlPlacementTop
Out[]=
​
Tangent unitary vector:
Value (​
t
1
​)
Point coordinates r[t]
Point
{6,0.99955,-0.119968}
Tangent unitary vector T[t]
TangentUnitaryVector
{0.,0.00750467,0.999972}