Definition:Cartesian 3-Space/Definition by Planes

Definition


Every point in ordinary $3$-space can be identified uniquely by means of an ordered triple of real coordinates $\tuple {x, y, z}$, as follows:

Identify one distinct point in space as the origin $O$.

Let $3$ distinct planes be constructed through $O$ such that all are perpendicular.

Each pair of these $3$ planes intersect in a straight line that passes through $O$.

Let $X$, $Y$ and $Z$ be points, other than $O$, one on each of these $3$ lines of intersection.

Then the lines $OX$, $OY$ and $OZ$ are named the $x$-axis, $y$-axis and $z$-axis respectively.

Select a point $P$ on the $x$-axis different from $O$.

Let $P$ be identified with the coordinate pair $\tuple {1, 0}$ in the $x$-$y$ plane.

Identify the point $P'$ on the $y$-axis such that $OP' = OP$.

Identify the point $P$ on the $z$-axis such that $OP = OP$.

The orientation of the $z$-axis is determined by the position of $P''$ relative to $O$.

It is conventional to locate $P''$ as follows.

Imagine being positioned, standing on the $x$-$y$ plane at $O$, and facing along the $x$-axis towards $P$, with $P'$ on the left.

Then $P$ is then one unit above'' the $x$-$y$ plane.

Let the $x$-$y$ plane be identified with the plane of the page or screen.

The orientation of the $z$-axis is then:


 * coming vertically "out of" the page or screen from the origin, the numbers on the $z$-axis are positive
 * going vertically "into" the page or screen from the origin, the numbers on the $z$-axis are negative.

Also defined as
Some sources do not specify that the $3$ constructed distinct planes need to be perpendicular.

If they are not, then what results is an oblique coordinate system.