Relative position of plane and plane

A distinction is made between three possible relative positions between two planes $E$ and $F$.

• They intersect.
  
• They are parallel.
  
• They are coincident.
  

Similar to the relative position of a line and a plane, one tries to calculate the intersection line.
However, if you come across a true statement (e.g. 0 = 0), the planes are coincident. If the statement is incorrect (e.g. 8 = 0), they are parallel.

Example

$\text{E: } x-y+z=2$

$\text{F: } 2x+y+z=4$

1. Set up a system of equations

   The two equations can be viewed as a system of equations.
   1. $x-y+z=2$
   2. $2x+y+z=4$

   Now you should eliminate a variable. This is achieved here, for example, by adding the two equations.

   I.+II.

   $3x+2z=6$

2. Replace variable with $r$

   One of the other variables is now replaced by $r$ and inserted in the equation. For example x:

   $\color{red}{x=r}$

   $3r+2z=6$

   The other variable ($z$) can now be expressed as a function of $r$. Simply solve equation for $z$.

   $3r+2z=6\quad|-3r$
   $2z=6-3r\quad|:2$
   $\color{red}{z=3-1.5r}$

   One of the two plane equations can also be used to determine $y$ by using $x$ and $z$.

   $x-y+z=2$
   $r-y+(3-1.5r)=2$
   $-0.5r-y+3=2\quad|+y$
   $-0.5r+3=2+y\quad|-2$
   $\color{red}{y=-0.5r+1}$

3. Set up equation of a line

   First we write the results for $x$, $y$ and $z$ among themselves.
   1. $x=r$
   2. $y=-0.5r+1$
   3. $z=3-1.5r$

   Sorted:

   1. $x=\color{blue}{0}\color{green}{+1}r$
   2. $y=\color{blue}{1}\color{green}{-0.5}r$
   3. $z=\color{blue}{3}\color{green}{-1.5}r$

   This can now be easily put into the form of a equation of a line.

   $\vec{x} = \begin{pmatrix} \, \\ \, \\ \, \end{pmatrix} + r \cdot \begin{pmatrix} \, \\ \, \\ \, \end{pmatrix}$

   $\vec{x} = \begin{pmatrix} \color{blue}{0} \\ \color{blue}{1} \\ \color{blue}{3} \end{pmatrix} + r \cdot \begin{pmatrix} \color{green}{1} \\ \color{green}{-0.5} \\ \color{green}{-1.5} \end{pmatrix}$

Example (parallel)

$\text{E: } x-y+z=2$

$\text{F: } 2x-2y+2z=7$

1. Set up a system of equations

   1. $x-y+z=2\,\,\,|\cdot(-2)$
   2. $2x-2y+2z=7$

   We apply the addition method.

   1. $-2x+2y-2z=-4$
   2. $2x-2y+2z=7$

   I.+II.
   $0=3$ f. s.

2. Interpret the result

   We obtain a contradiction or a false statement.

   $0\neq3$

   $E$ and $F$ therefore have no common point. The planes must be parallel.

   => $E$ and $F$ are parallel