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Real Plane
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Space Distance
Volume Measure
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Real Space


We are constantly thinking of $\R ^3$ as points of space.1


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.


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

  1. Future editions will modify this sheet. ↩︎
  2. Future editions will expand this. ↩︎
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