Math Planes Summary


Plane equation

A listing of the different plane equations:

Parametric equation
$\text{E: } \vec{x} = \vec{a} + r \cdot \vec{u} + s \cdot \vec{v}$
Vector equation
$\text{E: } (\vec{x} - \vec{a}) \cdot \vec{n}=0$
Cartesian equation
$\text{E: } ax+by+cz=d$
$\text{E: } \frac{x}a+\frac{y}b+\frac{z}c=1$

The last one is the intercept equation, a special form of the cartesian equation.

Converting plane equations

Plane equations
  1. parametric equation → vector equation

    Calculate normal equation (scalar product or cross product) and insert with position vector
  2. vector equation → parametric equation

    Find 2 direction vectors (perpendicular to the normal equation) and insert with position vector
  3. cartesian equation → vector equation

    Read off normal equation and search for any position vector in the plane
  4. normal equation → cartesian equation

    Multiply out (using the scalar product)
  5. cartesian equation → parametric equation

    Find any 3 points in the plane and set up parametric equation
  6. parametric equation → cartesian equation

    This method is hard so you should rather use the normal equation to get there (first 1, then 4).