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