# Thread: intersection of two planes.

1. ## intersection of two planes.

My geometry book says the intersection of two planes is a line, why can't the intersection of two planes be a point?

2. ## Re: intersection of two planes.

Hey hisajesh.

I'm assuming you mean in three dimensions (where the plane definition is ax + by + cz + d = 0 for some real numbers a,b,c,d).

The reason why it can't be a point is because the intersection of two planes (in 3D) can only share a common line, be the same plane, or be parallel (and have no intersection at all).

If you want the mathematical reason why then set up a system of equations using a matrix formulation of Ax = b where A is the [a b c] portion of the plane and b is the d portion of the plane equation and reduce the matrix.

You'll find that you have a 2 x 3 matrix which means that you will always at best have one free parameter. Because of this, you will always have (again at best) a line since a line is defined to have one free parameter.

The mathematical terms that describe this are known as rank and dimension (look them up for a clearer understanding) and the idea is that if you want a unique answer to a linear system then it means that Ax = b has a unique solution x if and only if A is square matrix (so 3x3 and not 2x3) and is also invertible (non-zero determinant).

3. ## Re: intersection of two planes.

just consider two sheets of cardboard, plywood or for that matter any material and visualize how would the intersect, it will be in a line. Sell the edge of a container it is a line segment and so on. Make observation around you and it would be quite clear.

4. ## Re: intersection of two planes.

Good idea idbutt.
Here are two planes of cardboard which are at right angles

They intersect at points all along the edge of the box
Also remember that the planes described by the equations extend to infinity which makes it impossible to arrange these cardboard sheets so that they only touch at 1 point.

5. ## Re: intersection of two planes.

Originally Posted by Shakarri
Good idea idbutt.
Here are two planes of cardboard which are at right angles

They intersect at points all along the edge of the box
Also remember that the planes described by the equations extend to infinity which makes it impossible to arrange these cardboard sheets so that they only touch at 1 point.
You are right recollect the definition of a line and line segment.