Coordinate Systems 02: three dimensional REALITY

Let's jump into defining of z -axis

Consider two straight lines X'OX and YOY' and these two lines are perpendicular to each other. Draw them on a plain graph paper.

Theße lines divide the plane into four regions - we call them as quadrants (derived from the word quater)

Division of coordinate plane by X'OX and YOY'

Now, draw a new and very different line ZOZ' which is such that it is perpendicular to both the previous lines. That our z-axis.

For z-axis we write it as ZOZ’ since ( Z'OZ : is not as per convention).

Now X'OX and YOY' (again we say : Y’OY is not as per convention as it is not diagrammatically true) divided the plane into four quadrant. Consider it like a chocolate cake with a layerings: Z-axis is that layerings seen from bottom.

It gives rise to eight equal parts, four quarters above having +Z axis; And, four quaters below having -Z axis. Each point on this 3D space is Cartesian product of Real numbers crossed Real no.s crossed Real numbers. It gives rise to ordered pairs with 3 enteries, that is (x, y, z)

Division of 3D space into eight octants (‘eight directions’) by ZOZ' & YOY' & X'OX

Eight octants

Earlier only two lines were there: so they could divide the space into 4 quadrants. Now additional z-axis or ZOZ' is like a layerings on cake : we are seeing the four quadrants from two sides (like seeing a cake from above and taking that to see that from below); this gives rise to division of space into 8 equal parts, called as octants.

Above view:

The octant that falls under zone of OX & YO & ZO is said First; it has all positive ordered pair of real numbers (x, y, +Z)

The octant under X'O and OY' and ZO is Third; it has all negative pairs, that is (-x, -y, +Z)

Second octant had (-x, y, +Z) and falls in zone of X'O and YO and ZO: it's inverse is Fourth octant that is (x, -y, +) and falls under OX and OY' and ZO

Bottom view:

The octant that falls under zone of OX & YO & OZ’ is said First; it has all positive ordered pair of real numbers (x, y, -Z)

The octant under X'O and OY’ and OZ’ is Third; it has all negative pairs, that is (-x, -y, -Z)

Second octant had (-x, y, -Z) and falls in zone of X'O and YO and OZ’; it'S JUST inverse is Fourth octant that is (x, -y, -Z) and falls under OX and OY' and OZ'