In the previous chapter we have discovered that it is possible to pinpoint any location on a plane with two numbers. The space we live in is not a flat plane though. It would be nice to have a coordinates system adequate for this situation too.
To figure it out, we will simply add a third dimension to the two-dimensional world described in the previous chapter and study the consequences.
As usually we will make some observations and perform some simple experiments to come up with a handful of claims from which everything else follows. Many things covered here may feel obvious and spending so much time proving them redundant. Please bear with me and treat this as an exercise in reasoning. It’s a good idea to refine our tools before we venture into areas where they will be needed. Let’s start our quest with studying the relations between basic objects in 3D.
Planes and segments in 3d space
If we grab a plane and some random point in space, this point must be on the plain or in space on either side of the plane. How can we tell whether two points are on the same or opposite side of the plain? A similar question was answered in 2d space so instead of reinventing the wheel, we will simply introduce a 3d version of postulate 3:
Postulate 3-3d: If a straight segment linking point P with Q intersects some plane k, then P and Q are on the opposite sides of k i.e. every route in space that links P and Q must also intersect plane k. If on the other hand the straight segment PQ doesn’t intersect k, both points are on the same side of plane k.
Can a straight line intersect any plane more than once? According to our definition of a plane if two different points are on the same plane then all points on the straight line linking them also belong to this plane. From this it follows that if a straight line has a common point with a plane then either it is the only common point or this line is on the plane.