\begin{definition}[Definition:Cartesian 3-Space/Definition by Axes]
600pxrightthumbA 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.
\end{definition}