Real Space
Why
We are constantly thinking of $\R ^3$ as points of space.1
Definition
We commonly associate elements of $\R ^3$ with points in space. (see Geometry).
There exists a set of all planes.
Let $P$ be the set of all planes of space.
Then $\cup P$ is the set of all lines and
$\cup \cup P$ is the set of all points.
There exists a one-to-one correspondence mapping
elements of $\cup \cup P$ onto elements of
$\R ^3$.
For this reason, we sometimes call elements of
$\R ^3$ points.
We call the point associated with $(0, 0, 0)$
the origin.
We call the element of $\R ^3$ which
corresponds to a point the
coordinates of the point.
Visualization
To visualize the correspondence we draw three perpendicular lines. We call these axes. We then associate a point of the line with $(0, 0, 0) \in \R ^3$. We can label it so. We then pick a unit length. And proceed as usual.2