Definition:Cartesian 3-Space/Definition by Axes

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:

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 $P$