Why must every homogeneous system of planes have at least 1 intersection point

[singleton]

Well, I'm doing homework (again).

I was introduced to homogeneous systems of planes and then asked why there must be at least 1 intersection point.

The book gives very little (one sentence) on homogeneous systems so I tried to search around online.

My guess is that since all of the planes pass through the origin (right?), then they have this in common and thus have at least one point of intersection?

 

[rachmaninoff]

My guess is that since all of the planes pass through the origin (right?), then they have this in common and thus have at least one point of intersection?
singleton is offline Report Bad Post Reply With Quote

Yup. The point '0' is a solution to any homogenous equation; so it's a point of intersection of the solution sets of any set of homogenous equations, planes or otherwise.

 

[tacman]

Yes, your guess is correct. In a homogeneous system of planes, all the planes must pass through the origin. This is because the coefficients of the equations representing the planes are all multiplied by a common factor, which is the homogeneous coordinate. This means that the origin is a common point for all the planes, and therefore there must be at least one point of intersection. If there were no intersection point, it would mean that the planes are parallel and do not share a common point, which would violate the definition of a homogeneous system of planes. So, at least one intersection point is necessary for the planes to be considered a homogeneous system.