Tangent Plane and Normal Vector

Question

I have this code that shows that the derivative is vertical to the surface. I need to change the point to an arrow that is vertical and moves as the point moves

f[x_, y_] = x + y - x y;
fx[x_, y_] = D[f[x, y], x];
fx[x_, y_] = Limit[(f[x + h, y] - f[x, y])/h, h -> 0];
fy[x_, y_] = D[f[x, y], y];
fy[x_, y_] = Limit[(f[x, y + h] - f[x, y])/h, h -> 0];
Manipulate[x0 = p[[1]]; y0 = p[[2]];
 Show[{Plot3D[{f[x, y], 
     f[x0, y0] + fx[x0, y0] (x - x0) + fy[x0, y0] (y - y0)}, {x, -5, 
     5}, {y, -5, 5}, BoxRatios -> {1, 1, 1}, 
    PlotRange -> {{-5, 5}, {-5, 5}, {-25, 25}}, 
    PlotStyle -> {Directive[Opacity[0.6]], 
      Directive[Orange, Opacity[0.6]]}, ViewPoint -> {2.5, -2, 1}, 
    ClippingStyle -> None], 
   Graphics3D[{PointSize[Large], 
     Point[{p[[1]], p[[2]], f[p[[1]], p[[2]]]}]}]}], {{p, {1, 
    1}}, {-1, -1}, {2, 2}}]

The change is so

I tried with the ِِArrow

Graphics3D[{Red, Arrowheads[0.1], 
  Arrow[Tube[{{1, 1, -1}, {2, 2, 0}, {3, 3, -1}, {4, 4, 0}}, 0.05]]}]

But I did not get the desired result.

Here here YouTube is aware of the way to use the Mathematica but did not display all the code.

Answer 1

For a lecture on differential geometry I did once a demonstration like you intend to (without a tangent plane though). May be it will be of some use to you. Here it is. I inserted the radius-vector that describes your surface. Please have a look.

 Manipulate[
 R0 = {xx, yy, 5 + (5 + xx + yy - xx*yy)/4};
 e1 = D[R0, xx];
 e2 = D[R0, yy];
 n = Cross[e1, e2]/Sqrt[e1.e1*e2.e2 - e1.e2^2];

 rule = {xx -> X, yy -> Y};
 Which[
  showR == False && showBasis == False && showN == False,
  Show[{
     Plot3D[R0[[3]] /. {xx -> x, yy -> y}, {x, -5, 5}, {y, -5, 5}, 
      PlotStyle -> Directive[Opacity[0.3]], Ticks -> None, 
      ViewPoint -> {13, -6, 5}, PlotRange -> All, Boxed -> False, 
      AxesOrigin -> {0, 0, 0}, ColorFunction -> "SunsetColors"],
     Graphics3D[{Text[Style["x", Italic, 16], {4.2, 0, 0.2}], 
       Text[Style["y", Italic, 16], {0.1, 2.4, 0.2}], 
       Text[Style["z", Italic, 16], {0, 0, 6.7}]}],
     Graphics3D[{PointSize[0.015], Point[R0], Point[{X, Y, 0}]}]
     }, BoxRatios -> Automatic] /. rule,

  showR == True && showBasis == False && showN == False,
  Show[{
     Plot3D[R0[[3]] /. {xx -> x, yy -> y}, {x, -5, 5}, {y, -5, 5}, 
      PlotStyle -> Directive[Opacity[0.3]], Ticks -> None, 
      ViewPoint -> {13, -6, 5}, PlotRange -> All, Boxed -> False, 
      AxesOrigin -> {0, 0, 0}, ColorFunction -> "SunsetColors"],
     Graphics3D[{Arrowheads[0.03], Thick, Blue, 
        Arrow[{{0, 0, 0}, R0}]}] /. rule,
     Graphics3D[{Text[Style["x", Italic, 16], {4.2, 0, 0.2}], 
       Text[Style["y", Italic, 16], {0.1, 2.4, 0.2}], 
       Text[Style["z", Italic, 16], {0, 0, 6.7}]}],
     Graphics3D[{Text[Style["R", Bold, Blue, 16], 
        Mean[{{0, 0, 0}, R0 + {1, 0, 0}}]], PointSize[0.015], 
       Point[R0], Point[{X, Y, 0}]}]
     }, BoxRatios -> Automatic] /. rule,

  showR == True && showBasis == True && showN == False,
  Show[{
     Plot3D[R0[[3]] /. {xx -> x, yy -> y}, {x, -5, 5}, {y, -5, 5}, 
      PlotStyle -> Directive[Opacity[0.3]], Ticks -> None, 
      ViewPoint -> {13, -6, 5}, PlotRange -> All, Boxed -> False, 
      AxesOrigin -> {0, 0, 0}, ColorFunction -> "SunsetColors"],

     Graphics3D[{Arrowheads[0.03], Thick, Blue, 
        Arrow[{{0, 0, 0}, R0}]}] /. rule,

     Graphics3D[{Text[Style["x", Italic, 16], {4.2, 0, 0.2}], 
       Text[Style["y", Italic, 16], {0.1, 2.4, 0.2}], 
       Text[Style["z", Italic, 16], {0, 0, 6.7}]}],

     Graphics3D[{Arrowheads[0.03], Thick, Red, 
        Arrow[{R0, R0 + Normalize[e1]}],
        Arrowheads[0.03], Arrow[{R0, (R0 + Normalize[e2])}]
        }] /. rule,

     Graphics3D[{Text[
        Style["\!\(\*SubscriptBox[\(e\), \(1\)]\)", Bold, Red, 
         16], {R0 + Normalize[e1] + {0.2, 0, 0}}], 
       Text[Style["\!\(\*SubscriptBox[\(e\), \(2\)]\)", Bold, Red, 
         16], {R0 + Normalize[e2] + {0, 0.2, 0}}], Red, 
       Text[Style["R", Bold, Blue, 16], 
        Mean[{{0, 0, 0}, R0 + {1, 0, 0}}]], PointSize[0.015], 
       Point[R0], Point[{X, Y, 0}]}]
     }, BoxRatios -> Automatic] /. rule,

  showR == True && showBasis == True && showN == True,
  Show[{
     Plot3D[R0[[3]] /. {xx -> x, yy -> y}, {x, -5, 5}, {y, -5, 5}, 
      PlotStyle -> Directive[Opacity[0.3]], Ticks -> None, 
      ViewPoint -> {13, -6, 5}, PlotRange -> All, Boxed -> False, 
      AxesOrigin -> {0, 0, 0}, ColorFunction -> "SunsetColors"],

     Graphics3D[{Arrowheads[0.03], Thick, Blue, 
        Arrow[{{0, 0, 0}, R0}]}] /. rule,

     Graphics3D[{Text[Style["x", Italic, 16], {4.2, 0, 0.2}], 
       Text[Style["y", Italic, 16], {0.1, 2.4, 0.2}], 
       Text[Style["z", Italic, 16], {0, 0, 6.7}]}],

     Graphics3D[{Arrowheads[0.03], Thick, Red, 
       Arrow[{R0, R0 + Normalize[e1]}],
       Arrowheads[0.03], Arrow[{R0, (R0 + Normalize[e2])}],
       Arrowheads[0.03], Darker@Green, Arrow[{R0, R0 + n}]
       }],

     Graphics3D[{Text[
        Style["\!\(\*SubscriptBox[\(e\), \(1\)]\)", Bold, Red, 
         16], {R0 + Normalize[e1] + {0.2, 0, 0}}], 
       Text[Style["\!\(\*SubscriptBox[\(e\), \(2\)]\)", Bold, Red, 
         16], {R0 + Normalize[e2] + {0, 0.2, 0}}], 
       Text[Style["n", Bold, Darker@Green, 
         16], {R0 + n + {0, 0, 0.2}}], 
       Text[Style["R", Bold, Blue, 16], 
        Mean[{{0, 0, 0}, R0 + {1, 0, 0}}]], PointSize[0.015], 
       Point[R0], Point[{X, Y, 0}]}]



     }, BoxRatios -> Automatic] /. rule


  ],

 Column[{
   Row[{Control[{{X, 1.7}, 0, 4}], Spacer[30], 
     Control[{{Y, 0}, -2, 2}]}],
   Row[{
     Control[{{showR, False}, {True, False}}], Spacer[30], 
     Control[{{showBasis, False}, {True, False}}], Spacer[30],
     Control[{{showN, False}, {True, False}}]}]

   }, Alignment -> Center], 
 ControlType -> {Slider, Slider, Checkbox, Checkbox, Checkbox}

    SaveDefinitions -> True]

To see the vectors check the check boxes. The green arrow shows the unit normal vector, the red ones the tangent unit vectors. It should look like the following:

Have fun!