# Thread: The Intersection of three planes

1. ## The Intersection of three planes

For what values of k will the following set of planes intersect in a line

x-2y-z=0
x+9y-5z=0
kx-y+z=0

Would like a step by step process. Please and Thank You.

2. Originally Posted by johntuan
For what values of k will the following set of planes intersect in a line

x-2y-z=0
x+9y-5z=0
kx-y+z=0

Would like a step by step process. Please and Thank You.
I can begin by saying that if the determinant of that system is 0 then there are infinitely many solutions because this is a homogenous system.

Now three planes can intersect in: a point (unique solution), a line, a plane (all three lie on top of eachother), pair of lines (two parrell planes), or no intersection (all three are parallel).

It should be clear that they must intersect (at (0,0,0)). It also seams that it cannot possibly be any of the other possibilities. So it is safe just to make the determinant zero:
$\displaystyle \left| \begin{array}{ccc}1&-2&-1\\1&9&-5\\k&-1&1\end{array} \right| = 0$.
Now solve.

3. actually im still sorta stuck what would I do from there?

4. Originally Posted by johntuan
actually im still sorta stuck what would I do from there?
Did you exapand the determinant?

5. have no idea what that means.

6. Originally Posted by johntuan
have no idea what that means.
Well then, do you know how to solve a system of three equations in three unknowns?
If so, the proceed!

If you do not understand either method then pray tell why were you asked to do this problem?