Section Summary: 13.5 (part b - planes in space)

1. Definitions

   The parametric equation of a plane P is

   

   The vector equation of a plane P is

   

   or

   

   Suppose that , , ; then

   

   This is the scalar equation of the plane P with normal vector n.

   These equations are not unique to a plane: any vector and vector normal to the plane will give another set of equations.

   By collecting terms in the scalar equation above we find that

   

   where This is called a linear equation in x, y, and z.

   Two planes are parallel if their normal vectors are parallel.

2. Theorems

   None to speak of.

3. Properties/Tricks/Hints/Etc.

   distinct planes in 3-space (or higher) can pair up in only two ways:

   • parallel
   • intersecting in a line
   The line of intersection can be found by solving both scalar equations of the planes simultaneously.

4. Summary

   In the second part of section 13.5, we are introduced to various ways of thinking about planes in space. We meet with several different equations of planes.