# How to tell if three planes are parallel or not

• Jun 3rd 2010, 10:17 PM
acevipa
How to tell if three planes are parallel or not
If I have three planes:

$\displaystyle a+3b+2c=5$
$\displaystyle 2a+b-c=2$
$\displaystyle 7a+11b+4c=13$

How do you determine if the planes are parallel.

And if they're not, how do you work out the line of intersection?

Thanks
• Jun 3rd 2010, 10:53 PM
earboth
Quote:

Originally Posted by acevipa
If I have three planes:

$\displaystyle a+3b+2c=5$
$\displaystyle 2a+b-c=2$
$\displaystyle 7a+11b+4c=13$

How do you determine if the planes are parallel.

And if they're not, how do you work out the line of intersection?

Thanks

1. Two planes are parallel if their normal vectors are collinear.

With your example $\displaystyle P_1 \nparallel P_2$ because there doesn't exist a $\displaystyle r \in \mathbb{R}$ such that

$\displaystyle \begin{pmatrix}1 \\ 3 \\ 2 \end{pmatrix} = r \cdot \begin{pmatrix}2\\1\\-1\end{pmatrix}$

2. To get the line of intersection: $\displaystyle P_1 \cap P_2$:

$\displaystyle \left|\begin{array}{rcl}a+3b+2c&=&5\\2a+b-c&=&2\end{array} \right.$ ........ $\displaystyle \implies$ ........ $\displaystyle 5a+5b=9 ~\implies~b=-a+\frac95$

Now set a = t. Then you get:

$\displaystyle \begin{pmatrix}a\\b\\c\end{pmatrix}=\begin{pmatrix }t\\-t+\frac95\\t-\frac15\end{pmatrix}$ which can be simplified to:

$\displaystyle \begin{pmatrix}a\\b\\c\end{pmatrix}= \begin{pmatrix}0\\\frac95\\-\frac15\end{pmatrix} + t \cdot \begin{pmatrix}1\\-1\\1\end{pmatrix}$

which is the equation of the line of intersection.

3. The next 2 lines should be calculated in just the same way. Have fun!

EDIT: The attached sketch shows the special cases:
• Jun 4th 2010, 05:00 AM
HallsofIvy
Quote:

Originally Posted by acevipa
If I have three planes:

$\displaystyle a+3b+2c=5$
$\displaystyle 2a+b-c=2$
$\displaystyle 7a+11b+4c=13$

How do you determine if the planes are parallel.

And if they're not, how do you work out the line of intersection?

Thanks

Two planes are parallel if and only if their "normal vectors" are parallel. Two planes, if not parallel, will intersect in a straight line.

Three planes are parallel if and onlyif they are parallel in pairs- which means, again, if and only if all three normal vectors are parallel. You can determine whether or not two vectors are parallel by dividing the corresponding components. They are parallel if and only if all three fractions are the equal.

Three planes typically intersect in a point not a plane. It may happen that the "line of intersection" of the first two planes is the same as the line of intersection of the last two- in that special case, all three planes intersect in a line. To find the line or point of intersection, attempt to solve the three equations simultaneously. If you can solve for all of a, b, and c, that is the point of intersection. If it turns out that after reducing from three equations in three variables to two equations in two variables, the two equations are the same, you can select either of the remaining variables as a "parameter" and solve for the other two in terms of that parameter. That will give parametric equations for the "line of intersection".

But, again, three planes intersecting in a line is a very unusual situation.