Originally Posted by

**HallsofIvy** I'm not sure I understand your question. You said the "intersection of two hyperplanes". A hyperplane in n dimensions is a linear manifold of dimension n- 1, not just any hyperplane. In particular, any hyperplane in $\displaystyle R^3$ is a plane and two planes (unless parallel) intersect in a line. A line is NOT a hyperplane in $\displaystyle R^3$ and, in general, two lines do NOT intersect in $\displaystyle R^3$.

If, by "hyperplane", you mean any linear manifold, and you are working in, say, n dimensions and want to find the intersection between a 2 dimensional linear manifold and a 4 dimensional manifold, then about the best you can do is put the equations for one into the equations for the other and try to solve. Of course, it might well happen that they do NOT intersect.

In n dimensions, you can write a 2 dimensional linear manifold as n linear equations in two parameters. You can write a 4 dimensional manifold as n linear equations in 4 parameters. Setting the coordinates equal, you will have n linear equations in 4+ 2= 6 parameters. If n= 6 then you can expect to solve those for one point of intersection. If n< 6, you will have fewer equations than parameters and will have to solve for 6- n of them in terms of the other n. That is, the intersection will be a 6- n dimensional linear subset of the 4 dimensional manifold (of course n must be greater than or equal to 4 so 6- n is less than or equal to 2). If n> 6 you will have more equations than parameters and, in general, the two manifolds do not intersect.