Definition:Cartesian 3-Space/Definition by Axes

Definition

A general point in Cartesian $3$-Space

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:

Construct a Cartesian plane, with origin $O$ and axes identified as the $x$-axis and $y$-axis.

Recall the identification of the point $P$ with the coordinate pair $\tuple {1, 0}$ in the $x$-$y$ plane.

Construct an infinite straight line through $O$ perpendicular to both the $x$-axis and the$y$-axis and call it the $z$-axis.

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

Identify the $z$-axis with the real number line such that:

$0$ is identified with the origin $O$
$1$ is identified with the point $P$

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

It is conventional to use the right-handed orientation, by which we 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.

Cartesian Coordinate Triple

Let $Q$ be a point in Cartesian $3$-space.

Construct $3$ straight lines through $Q$:

one perpendicular to the $y$-$z$ plane, intersecting the $y$-$z$ plane at the point $x$
one perpendicular to the $x$-$z$ plane, intersecting the $x$-$z$ plane at the point $y$
one perpendicular to the $x$-$y$ plane, intersecting the $x$-$y$ plane at the point $z$.

The point $Q$ is then uniquely identified by the ordered pair $\tuple {x, y, z}$.