Maximiliano,
to be able to see the orthogonality you basically need (in Plot3D and friends) the option BoxRatios -> Automatic to make sure that every axis has the same scale - otherwise things look quite odd. Just for completeness here some code (I use Gradient and InfiniteLine):
ClearAll["Global`*"]
ellps[x_, y_, z_] := x^2/4 + y^2 + z^2/9
sol = SolveValues[ellps[x, y, z] == 3, z];
grad[x_, y_, z_] = Grad[ellps[x, y, z], {x, y, z}];
plane[x_, y_] := -(3/2) x + 3 y - 9 (*Tangent plane to the point (-2,1,-3)*)
point = {-2, 1, -3};
Show[Plot3D[{plane[x, y], First[sol]}, {x, -3.5, 3.5}, {y, -2, 2},
PlotRange -> {-7, 2}, BoxRatios -> Automatic],
Graphics3D[{Thickness[.01], InfiniteLine[point, grad @@ point]}]]
Greetings to Santiago - nearly 30 years ago I spent 6 month at your campus!
Henrik
