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
parametric equation → vector equation
Calculate normal equation (scalar product or cross product) and insert with position vectorvector equation → parametric equation
Find 2 direction vectors (perpendicular to the normal equation) and insert with position vectorcartesian equation → vector equation
Read off normal equation and search for any position vector in the planenormal equation → cartesian equation
Multiply out (using the scalar product)cartesian equation → parametric equation
Find any 3 points in the plane and set up parametric equationparametric equation → cartesian equation
This method is hard so you should rather use the normal equation to get there (first 1, then 4).