Intersection of 3d space with another 3d space. (is this nonsense?) I'm reading Basic Mathematics from Serge Lang and reached to the point with isometries in 3d space (that's in the exercise section). It got me thinking.
The question:
If the intersection of two lines(in 2d space) is point and the intersection of two planes (in 3d space) is line, is there such thing as intersection of two 3d spaces? 
Thanks in advance for everyone who reply! And please take in account that I'm reading Basic Mathematics(emphasis on basic). Answer like "google these things..." will be satisfying enough. 
 A: You started with the depiction of two lines in a plane intersecting in a point.  It is also possible that two distinct lines in a plane have no intersection, i.e. the lines are parallel.
In the case of two distinct planes in three dimensional space we have the following possibilities for their intersection:


*

*They intersect in a common line.  

*They do not intersect (parallel planes).


If we put two distinct planes (two dimensions) inside a four dimensional space, then in addition to the above we have the possibility:


*

*They intersect in a common point.


Let's use the notation of Cartesian coordinates to explain this.  Suppose we attach real coordinates $(x,y,z,t)$ to four dimensional space.  One two dimensional subspace ("plane") consists of points $\{(x,y,0,0)\mid x,y\in \mathbb R\}$, and another (for instance) consists of points $\{(0,0,z,t)\mid z,t\in \mathbb R\}$.  The only common point is the origin $(0,0,0,0)$.
In like fashion we can consider three dimensional subspaces of four (or higher) dimensional space.  As the OP contemplated in a Comment above, taking $U = \{(x,y,z,0)\mid x,y,z\in \mathbb R\}$ and $V = \{(0,y,z,t)\mid y,z,t\in \mathbb R\}$ gives a two dimensional intersection:
$$ U \cap V = \{(0,y,z,0)\mid y,z\in \mathbb R\} $$
In addition to "Euclidean geometry" of various kinds (plane, solid, etc.), we use the term synthetic geometry to refer to studies where Cartesian coordinates are attached to points, allowing algebra to inform our solution of geometric problems.  Such studies are customarily introduced in secondary school education and continue in college and beyond.
A related topic in undergraduate college courses is linear algebra.  Courses in analysis may advance the studies from finite-dimensional spaces to their infinite-dimensional "cousins".
