How to find the plane at which two hyperplanes intersect.

[jenny_shoars]

I know that to find the line at which two planes intersect, you can take the cross product of their normal vectors. This gives you a vector parallel to the line. Then you can just find a point which lies on both planes and that position plus the vector is your line.

How do you do the equivalent for the plane at which two hyperplanes intersect? I would initially think you could do something like take the determinant of the two hyperplanes in the same way that the cross product is from the determinant of the the two planes. However, the two hyperplanes don't give a square matrix the same way the two regular planes do. Also, how would you go about finding a point which lies on both hyperplanes in order to get the fully determined plane?

Thank you for your time!

 

[homeomorphic]

One way to look at it is that it's just a linear algebra problem. The first plane is a set that satisfies some linear equations, so is the second plane and the intersection is the set the satisfies both sets of equations.

Cross products don't really exist in higher dimensions. An appropriate analog would be the wedge product, which could also be used to find the intersection.

By the way, hyperplane usually means one dimension less than the ambient space, not just higher dimensional planes.

 

[jenny_shoars]

Of course. You're right on both accounts. Thank you much!