Definition:Cartesian 3-Space

Definition


Every point in ordinary $3$-space can be identified uniquely by means of an ordered triple $\tuple {x, y, z}$ of real coordinates.

Two perpendicular straight lines are chosen, and then a third straight line perpendicular to those two.

These are called the axes.

They are understood to be infinite.

The usual directions to make them are:
 * $(1): \quad$ Across the page, from left to right. This is usually called the $x$-axis.
 * $(2): \quad$ Up the page, from bottom to top. This is usually called the $y$-axis.
 * $(3): \quad$ Out of the page, from behind to forward. This is usually called the $z$-axis.

The point of intersection of the axes is called the origin.

A unit length is specified.

The axes are each identified with the set of real numbers $\R$, where the origin is identified with $0$.

The real numbers increase to the right on the $x$-axis, upwards on the $y$-axis, and out of the page on the $z$-axis.

Thus:
 * to the left of the origin the numbers on the $x$-axis are negative
 * below the origin the numbers on the $y$-axis are likewise negative.
 * behind the origin the numbers on the $z$-axis are likewise negative.

Thus space can be identified with the cartesian product $\R^3$.

In this context, $\R^3$ is called the (cartesian) coordinate 3-space.