How do I find the plane at which two hyperplanes intersect?

Question

In the case of finding the line at which two planes intersect, you need to take the cross product of the normal of the two planes. This cross product is simply taking the determinant of matrix:

i  j  k
x1 y1 z1
x2 y2 z2

Where (x, y, z) is the normal vector of each plane. The result is a vector parallel to the intersection line. From there you need to find a point which lies on both planes. The two parts combined give you a fully defined line.

How can this be extended to hyperplanes intersecting at a plane? I would assume I would need to take the determinant of a similar matrix, but the matrix I think of:

h  i  j  k
w1 x1 y1 z1
w2 x2 y2 z2

Is not a square matrix. Also, I don't know how to find a point which lies on both hyperplanes.

Can anyone explain to me how to find the intersection plane of the hyperplanes?

Thanks for your time!

Answer 1

You don't have to calculate a determinant for that, just perform a simple variable replacement and you will get the intersection plane. For instance, if you have two hyperplanes:

3x + 4y + 2z - 7w = 10
2x - 3y + 2z + 1w = 2

You can then isolate w (or any other variable):

w = 2 - 2x + 3y - 2z

And replace it in the first equation:

3x + 4y + 2z - 7(2 - 2x + 3y - 2z) = 10

Which results in:

17x - 17y + 16z - 14 = 10

And now you have your intersection plane. Just simple math.

The complete 4D plane representation is based on both equations, first you find (x, y, z) values that solve 17x - 17y + 16z - 14 = 10 and then you calculate w using w = 2 - 2x + 3y - 2z.