# Find intersection of three planes

I am studying and my friends and I are stuck on one question that has us all thinking.

It goes something like this:

Give an example of three planes that only intersect at $(x, y, z) = (1,2,1)$. Justify your choice. The three planes form a linear system of equations.

Solve this by using Cramer's rule.

If you have time, my friends and I would like a motivated answer :)

• Can you tell us what you have tried and your thoughts on the matter? – Amzoti Mar 27 '13 at 23:14

## 1 Answer

A simple answer to this would be the following set of planes:

$x=1$

$y=2$

$z=1$

Though this doesn't use Cramer's rule, it wouldn't be that hard to note that these equations would form the Identity matrix for the coefficients and thus has a determinant of 1 and would be solvable in a trivial manner.

One could create more elaborate plan equations by taking this system and doing various linear transformations on those equations. Such a result could be:

$x+y+z=4$ (Sum of all 3)

$x-y+z=0$ (First and third minus second)

$x+y-z=2$ (First and second minus third)

That would have a different determinant and thus have the same set of solutions though now it looks a bit more complicated.