Plotting normal vectors to planes and to paraboloids (picture seems inaccurate)

Question

I have a somewhat complicated picture (actually two) to plot that I've been struggling with in Mathematica... the first one almost worked:

p1 = Plot3D[{x + y, x - y, -x + 2 y}, {x, -10, 10}, {y, -10, 10}, 
  PlotStyle -> {{Green, Opacity[0.5]}, {Yellow, Opacity[0.35]}, {Blue,
      Opacity[0.2]}}, Mesh -> None, Boxed -> False, Axes -> True, 
  BoxRatios -> {1, 1, 1}, AxesOrigin -> {0, 0, 0}, 
  AxesStyle -> Opacity[0]]
p2 = Graphics3D[{{Blue, Arrow[{{0, 0, 0}, {-10, 20, -10}}]}, {Red, 
    Arrow[{{0, 0, 0}, {10, 10, -10}}]}, {Blue, 
    Arrow[{{0, 0, 0}, {10, -10, -10}}]}}, BoxRatios -> {1, 1, 1}, 
  Axes -> False, AxesLabel -> {"x", "y", "z"}, 
  AxesStyle -> RGBColor[0, 0, 0], BaseStyle -> 12]
Show[p1, p2]

It is supposed to show three planes and their normal vectors at the origin, but they don't look perpendicular to the planes... is there a way to fix this?

The other picture is similar, but with the paraboloid y=|x|^2 instead of planes. I would like to plot three normal vectors at different points, but is there a way to do that without having to compute the vectors manually from the equation of the surface? Any inputs are welcome... :)

Edit: If one uses this code for the paraboloid (based on the answer given below), only one of the pieces shows up. Is there a fix for this?

w1 = Plot3D[x^2 + y^2, {x, -5, 5}, {y, -5, 5}, 
   PlotStyle -> {Green, Opacity[0.5]}, Mesh -> None, Boxed -> False, 
   Axes -> True, AxesOrigin -> {0, 0, 0}, AxesStyle -> Opacity[0], 
   BoxRatios -> Automatic];
w2 = Plot3D[x^2 + y^2, {x, 10, 15}, {y, 10, 15}, 
   PlotStyle -> {Green, Opacity[0.5]}, Mesh -> None, Boxed -> False, 
   Axes -> True, AxesOrigin -> {0, 0, 0}, AxesStyle -> Opacity[0], 
   BoxRatios -> Automatic];
Show[w1, w2]

Answer 1

If you remove AxesStyle -> Opacity[0], you can see that the $z$ axis is shown from approximately -30 to 30. Therefore, the image is squeezed (because your forced the BoxRatios to be {1, 1, 1}), and that is why your normal vector do not look perpendicular (even though they are).

Solution 1

Use PlotRange -> {-10, 10} to fix the range of $z$ axis and then ClippingStyle -> None to remove the gray part outside the plotting range.

p1 = Plot3D[{x + y, x - y, -x + 2 y}, {x, -10, 10}, {y, -10, 10}, 
   PlotStyle -> {{Green, Opacity[0.5]}, {Yellow, 
      Opacity[0.35]}, {Blue, Opacity[0.2]}}, Mesh -> None, 
   Boxed -> False, Axes -> True, BoxRatios -> {1, 1, 1}, 
   AxesOrigin -> {0, 0, 0}, AxesStyle -> Opacity[0], 
   PlotRange -> {-10, 10}, ClippingStyle -> None];
p2 = Graphics3D[{{Blue, Arrow[{{0, 0, 0}, {-5, 10, -5}}]}, {Green, 
     Arrow[{{0, 0, 0}, {5, 5, -5}}]}, {Yellow, 
     Arrow[{{0, 0, 0}, {5, -5, -5}}]}}, BoxRatios -> {1, 1, 1}, 
   Axes -> False, AxesLabel -> {"x", "y", "z"}, 
   AxesStyle -> RGBColor[0, 0, 0], BaseStyle -> 12];
Show[p1, p2]

Note: I have changed the colors of the arrows to match the planes and divided their length by 2, so that they all fit into your range (-10 to 10).

Solution 2

As mentioned by @Simon in the comments, you can also set BoxRatios -> Automatic. In this case, no clipping is needed and planes are displayed as tetragons.

p1 = Plot3D[{x + y, x - y, -x + 2 y}, {x, -10, 10}, {y, -10, 10}, 
   PlotStyle -> {{Green, Opacity[0.5]}, {Yellow, 
      Opacity[0.35]}, {Blue, Opacity[0.2]}}, Mesh -> None, 
   Boxed -> False, Axes -> True, AxesOrigin -> {0, 0, 0}, 
   AxesStyle -> Opacity[0], BoxRatios -> Automatic];
p2 = Graphics3D[{{Blue, Arrow[{{0, 0, 0}, {-5, 10, -5}}]}, {Green, 
     Arrow[{{0, 0, 0}, {5, 5, -5}}]}, {Yellow, 
     Arrow[{{0, 0, 0}, {5, -5, -5}}]}}, Axes -> False, 
   AxesLabel -> {"x", "y", "z"}, AxesStyle -> RGBColor[0, 0, 0], 
   BaseStyle -> 12];
Show[p1, p2]