Formalizing ideas of intersecting spaces What can a plane and line form when they intersect (e.g. a line? a point? a plane?).
What can a plane and another plane form when they intersect?
How can I answer such questions systematically, and in the general case for $\mathbb R^n$?
 A: It is not entirely clear from your question what type of geometric objects you are interested in. From the examples you cite it would seem that you are interested in affine subspaces of $\mathbb R^n$. An affine subspace is a translation of a linear subspace of $\mathbb R^n$. Linear subspaces in $\mathbb R^n$ correspond, dimension-wise, to the origin, a straight line through the origin, a plane through the origin, hyperplanes through the origin, and so on. So, affine subspaces correspond to arbitrary points, arbitrary straight lines, arbitrary planes, and so on. 
Life in the the realm of linear subspaces is rather simple. Every subspaces has a well-defined dimension and the dimension determines the possible behaviour of intersections. A fundmental equation relating various dimensions is $dim(V + W)+dim(V\cap W)=dim(V) + dim(W)$. So, if $V$ and $W$ are straight lines through the origin, then they both have dimension $1$. $V+W$ is then either a plane (if the two lines are different) or a line again, in any case $dim(V+W)$ is either $1$ or $2$, and thus the dimension of the intersection is either $0$ (and is thus a point) or $1$, and is thus a line. So, the intersection of two straight lines through the origin is either the origin (if the two lines are different) or it is $V$ if $V=W$. When higher dimensions are involved there are more possibilities but the analysis is similar. 
The affine case is slightly more complicated. The intersection of two straight lines may be empty. Still, analyzing affine subspaces still benefits a lot from the simplicity of the linear case. 
Far more complicated are manifolds in $\mathbb R^n$. A manifold of dimension $d$ is a geometric subspace of $R^n$ that locally looks like a piece of $\mathbb R^d$. Here things become much more complicated. Differential topology/geometry is the subject dealing with such issues. The classification of manifolds according to dimensions is very hard. In the case of one dimensional manifolds for instance, it can be shown that a manifold is either essentially the same as a (finite or infinite) line, or it is essentially the same as the circle. In higher dimension there are many more possibilities (e.g., the ). However, if you care about how the manifolds are embedded in the ambient space, then this leads to more complicated things. For instance, in dimension one this leads to knot theory. 
