In order to give an explicit description of the intersection of two hyperplanes, H1 and H2, in R^n, we need to find a point p in H1 N H2 and two linearly independent vectors vi and v2 that span a subspace parallel to the 2-dimensional flat H1 N H2.
Given that H1 is the hyperplane through point p1 with normal vector n1, and H2 is the hyperplane through point p2 with normal vector n2, we can begin by finding the equation of each hyperplane.
The equation of a hyperplane in R^n is given by:
n • (x – p) = 0
where n is the normal vector of the hyperplane, p is any point on the hyperplane, and • denotes the dot product.
For H1, we have the equation:
n1 • (x – p1) = 0
Similarly, for H2, we have the equation:
n2 • (x – p2) = 0
To find the intersection of these two hyperplanes, we need to find a point x that satisfies both equations simultaneously. This means that x lies in both H1 and H2, which gives us the condition H1 N H2.
Let’s solve for x:
n1 • (x – p1) = 0
n2 • (x – p2) = 0
Expanding these equations, we get:
n1 • x – n1 • p1 = 0
n2 • x – n2 • p2 = 0
Now, let’s express x in terms of p1 and p2:
n1 • x = n1 • p1
n2 • x = n2 • p2
This gives us a system of linear equations:
n1 • x – n1 • p1 = 0
n2 • x – n2 • p2 = 0
We can rewrite this system of equations as a matrix equation:
A • x = b
where A is a matrix whose rows are the normal vectors of H1 and H2 (A = [n1; n2]), x is the vector x = [x1; x2; … ; xn], and b is the vector b = [n1 • p1; n2 • p2].
To find the point p in the intersection of H1 and H2, we need to solve this system of equations.
Now, let’s find two linearly independent vectors vi and v2 that span a subspace parallel to the 2-dimensional flat H1 N H2.
Since H1 N H2 is a 2-dimensional flat, we know that it can be spanned by two linearly independent vectors. Let’s call these vectors vi and v2.
To find vi and v2, we can use the fact that they are orthogonal to the normal vectors n1 and n2, respectively. This means that vi • n1 = 0 and v2 • n2 = 0.
By solving these equations, we can find two linearly independent vectors vi and v2 that span the desired subspace.
Once we have found p and the vectors vi and v2, we can provide an explicit description of H1 N H2 by stating the equations of the hyperplanes, the point p, and the span of the vectors vi and v2.
Overall, finding the intersection of two hyperplanes in R^n involves solving a system of linear equations and finding linearly independent vectors that span the desired subspace.
