Math Relative position of planes Plane and plane

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.


If two planes intersect, there is no point of intersection but a line intersection.

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.



The easiest way to calculate the intersection line is when both planes are in the cartesian form.


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

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

  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.



  2. Replace variable with $r$

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



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


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


  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. $z=3-1.5r$


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

    $0=3$ f. s.

  2. Interpret the result

    We obtain a contradiction or a false statement.


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

    => $E$ and $F$ are parallel



Two parallel planes can also be recognized by the fact that the normal vectors of the planes are multiples of one another (collinear).