Relative position of three planes

To study the relative position of three planes $\pi_1(A;\overrightarrow{u}, \overrightarrow{v}), \pi_2(A'; \overrightarrow{u'},\overrightarrow{v'})$ and $\pi_3(A''; \overrightarrow{u''}, \overrightarrow{v''})$ expressed by their general equations:

$$\begin{array}{rrcl} \pi_1:&A_1x+B_1y+C_1z+D_1&=&0 \\ \pi_2:& A_2x+B_2y+C_2z+D_2&=&0 \\ \pi_3:&A_3x+B_3y+C_3z+D_3&=&0\end{array}$$

Let's consider the system formed by three equations. The matrix $M$ and $M'$ associated with the system are:

$$M=\begin{pmatrix} A_1 & B_1 & C_1 \\ A_2 & B_2 & C_2 \\ A_3 & B_3 & C_3 \end{pmatrix}$$

$$M'=\begin{pmatrix} A_1 & B_1 & C_1&-D_1 \\ A_2 & B_2 & C_2 &-D_2\\ A_3 & B_3 & C_3&-D_3 \end{pmatrix}$$

We can classify the relative position of the planes by the compatibility of the systems:

Compatible system

Indeterminate Compatible system

$$rank (M) = rank (M') = 1$$

The solutions depend on two parameters. Three planes are equal.

$$rank (M) = rank (M') = 2$$

The solutions depend on a parameter, therefore they have a straight line in common.

Now we must determine the position of the planes two by two. We have 2 options:

• Three planes are cutting in a straight line.
• Two planes are equal and cut the other plane.

Determinate compatible system

$$rank (M) = rank (M') = 3$$

The planes are cutting at a point.

Incompatible system

$rank (M) = 1$; $rank (M') = 2$

Incompatible system: parallel planes.

Next we must determine if they coincide or not.

$rank (M) = 2$; $rank (M') = 3$

Incompatible system: they are cutting planes.

Next, it is necessary to determine if there are also parallel planes.