Math Planes Summary

# 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

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).