VectorScaling on VectorPlot produces vectors of wrong length

Question

I'm trying to plot the following VectorField:
VectorPlot[{x/Sqrt[x^2 + y^2], y/Sqrt[x^2 + y^2]}, {x, -5, 5}, {y, -5, 5}]

I want to use a VectorScaling-option such that vectors are displayed accoring to their length. Obviously, all vectors in this specific vector plot should have the same length, as they are normalized.
I tried to achieve this using the options
VectorScaling->Automatic

This yields the following output:

The vectors have different lengths (and the colors match this). Why does this happen? The options "Linear", "Log" and "Sqrt" yield similar results. Is this an artifact of calculation inaccuracies? If so, how may I compensate for this?
If I don't use any VectorScaling-option, then all the vectors have the same length (which is the default), but the colors still vary.
I'm using Wolfram Mathematica 12.3.
Addendum: I understand that getting rid of "VectorScaling->Automatic" would yield vectors of constant length. Though I'd like to plot three different vector fields in such a way, that the length of the plottet vectors relates to the real length. So I need some sort of vector scaling for this field of constant length, too.

Alrubaie · Answer

f1[x_, y_] := x/Sqrt[x^2 + y^2];
f2[x_, y_] := y/Sqrt[x^2 + y^2];

p1 = VectorPlot[{f1[x, y], f2[x, y]}, {x, -5, 5}, {y, -5, 5}, 
      VectorScale -> {Automatic, Automatic, #1 &}, ImageSize -> Medium]

p2 = VectorPlot[{f1[x, y], f2[x, y]}, {x, -5, 5}, {y, -5, 5}, 
  VectorScale -> {Automatic, Automatic, #2 &}, ImageSize -> Medium]

Finally
Show[p1, p2]

VectorScaling on VectorPlot produces vectors of wrong length

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