What algebraic structure is if a vector space is immersed on a intersection between plane-hyperplane?

Question

Usually we refer to a vector space immersed into a plane or an hyperplane. I want to understand what happens if I have this structure

• a vector or a vector space
• 2 different planes: a plane & hyperplane
• plane and hyperplane are intersected between them

I want to immerse vector space in this 2-different plane intersection.
What algebraic structure is ? What happens ?

Answer 1

Since you call them "2 different planes", I assume neither is a subset of the other. In that case, their intersection will either be a line (a one-dimensional vector space) or the origin (a zero-dimensional vector space). if you allow one to be a subset of the other, then obviously a plane is also a possibility.

Vector spaces are not "immersed in" a plane or hyper-plane. Instead they are almost exactly what a "hyper-plane" is. A hyper-plane is a geometric object with the property that given any two points in it, the entire line passing through those points is also contained in the hyper-plane. Because of this property, if you choose a point in the hyperplane to act as origin, then there are natural geometric constructions that define vector addition and scalar multiplication between all points of the hyper-plane, which makes the hyper-plane a vector space.