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Tangent Plane & Normal Line Calculator

Tangent-plane calculator. Enter a surface and a point to find the tangent plane equation and normal vector, with partials.

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z−z₀ = f_x(x−x₀)+f_y(y−y₀)
e.g. x^2+y^2

📖 Tutorial | Tangent Plane

1. Definition
The tangent plane to z=f(x,y) at (x₀,y₀,z₀) is z−z₀=f_x(x−x₀)+f_y(y−y₀); the normal vector is (f_x,f_y,−1).
2. Symbols
point(x₀,y₀,z₀)
f_x,f_ypartials
normal(f_x,f_y,−1)
3. How it works
  • Find z₀ and partials;
  • Substitute into the plane formula;
  • Rearrange as z=Ax+By+C.
4. Steps
  1. Enter z=f(x,y);
  2. Enter (x₀,y₀);
  3. Click Calculate.
5. Example
Example: z=x²+y² at (1,1,2).
Solution: f_x=f_y=2; plane z−2=2(x−1)+2(y−1), i.e. z=2x+2y−2.
6. Pitfalls
The plane touches at the point;
The normal is perpendicular to the plane;
The plane is the geometric form of the total differential.

❓ FAQ | Tangent Plane

Relation to differential?
The plane is the geometric form of df.
How to orient the normal?
By (f_x,f_y,−1).
Why z-component −1?
From F=z−f(x,y)=0.
Normal line?
(x−x₀)/f_x=(y−y₀)/f_y=(z−z₀)/(−1).
Implicit surface?
Solve for z=f(x,y) first.