Definition:Cartesian 3-Space/Ordered Triple

Identification of Point in Space with Ordered Triple
Every point in space can be identified by means of an ordered triple of real coordinates $\tuple {x, y, z}$, as follows:

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

Select a point $P$ in space different from $O$.

Let the distance from the origin to $P$ be defined as being $1$.

Draw an infinite straight line through $O$ and $P$ and call it the $x$-axis.

Select an infinite straight line through $O$ perpendicular to $OP$ and call it the $y$-axis.

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

Now, let $Q$ be any point in the space.

Draw $3$ lines through $Q$, parallel to the $x$-axis, $y$-axis and $z$-axis.

The $x$-$y$ plane is then conventionally oriented so that:
 * the $x$-axis is horizontal with $P$ being to the right of $O$.
 * the $y$-axis is perpendicular to the $x$-axis so that facing from the origin to $P$ along the $x$-axis, the $y$ coordinate increases to the left.

Then the $z$-axis is perpendicular to the page so that, standing on the $x$-$y$ plane, facing from the origin to $P$ along the $x$-axis, and with the $y$ coordinate increasing to the left, the $z$ coordinate increases upward.

Thus the point $Q$ can be uniquely identified by the ordered pair $\tuple {x, y, z}$ as follows:

$z$ Coordinate
The point $P$ is identified with the coordinates $\tuple {1, 0, 0}$.

Also known as
The ordered triple $\tuple {x, y, z}$ which determines the location of $P$ in the cartesian $3$-space can be referred to as the rectangular coordinates or (commonly) just coordinates of $P$.