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I have a function of two variables which returns a vector of three components. For example

func[l_,t_]:={
    x=....
    y=....
    z=....
    {x,y,z}}

Now I want to do a 3D parametric plot for this function with the parameter being t for different values of the variable l(say 10 values).

I do the following

ParametricPlot3D[Evaluate[Table[func[l, t][[1]], {l, 1,10}]], {t, 0, 20}]

This works fine and I get 10 different lines with 10 different colors.

What I want is that the color of the lines should be mappable to a colorscale so that color of the line corresponds to the value of the parameter l on the colorscale. I also want the color scale to appear next to the plot.

How to do this? Thanks

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Okay, so take some form for your function

func[l_, t_] := {
  x = 3 t Sin[l t];
   y = 3 t Cos[l t];
   z = l t + l; 
  {x, y, z}}

Then one way to get what you are looking for is like this,

Show[
 Table[
  ParametricPlot3D[
   func[l, t][[1]], {t, 0, 5}, PlotStyle -> Hue[l/10], 
   PlotLegends -> {"l = " <> IntegerString[l]}],
  {l, 1, 10}],
 PlotRange -> All]

enter image description here

Here the color of the line is a function of l (specifically, it is Hue[l/10]). If you want a color bar instead, use

BarLegend[{Hue, {0, 10}}, ColorFunctionScaling -> True]

enter image description here

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