How to find the plane at which two hyperplanes intersect.

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In summary, to find the line of intersection for two planes, you can take the cross product of their normal vectors and add it to a point that lies on both planes. However, for hyperplanes, the equivalent would be to use the wedge product instead of the cross product. The hyperplanes can also be seen as sets that satisfy linear equations, and their intersection would be the set that satisfies both sets of equations. The term "hyperplane" typically refers to a plane in a higher dimensional space, not just a regular higher dimensional plane.
  • #1
jenny_shoars
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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!
 
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  • #2
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.
 
  • #3
Of course. You're right on both accounts. Thank you much!
 

1. What is a hyperplane?

A hyperplane is a flat subspace in a higher-dimensional space. In 3-dimensional space, a hyperplane is a 2-dimensional flat surface, and in n-dimensional space, a hyperplane is an (n-1)-dimensional flat subspace.

2. How do you find the equation of a hyperplane?

The equation of a hyperplane can be found by using the normal vector and a point on the plane. The equation is in the form of Ax + By + Cz + ... = D, where A, B, C, etc. are the components of the normal vector and D is a constant.

3. What is the relationship between two intersecting hyperplanes?

When two hyperplanes intersect, they share a common point or line. This point or line is the solution to the system of equations formed by the equations of the two hyperplanes. If the two hyperplanes are parallel, there is no intersection and the system of equations has no solution.

4. How do you determine if two hyperplanes intersect?

Two hyperplanes intersect if their equations are not parallel. This can be determined by checking if the normal vectors of the two hyperplanes are parallel. If the normal vectors are not parallel, the hyperplanes intersect at a common point or line.

5. Can three or more hyperplanes intersect at a single point?

Yes, three or more hyperplanes can intersect at a single point. This point is the solution to the system of equations formed by the equations of the hyperplanes. In 3-dimensional space, three non-parallel planes can intersect at a single point.

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