Definition:Axis

General Definition
An axis is the name used for a general infinite straight line which is particularly significant in some particular way in the study of linear transformations of a Real Vector Space.

Coordinate Axes
Consider a coordinate system.

One of the reference lines of such a system is called an axis.

The plural of axis is axes (pronounced "ax-eez" rather than "ax-iz" - compare basis).

X-Axis
In a cartesian coordinate system, the x-axis is the one usually depicted and visualised as going from left to right.

It consists of all the points in the Real Vector Space in question (usually either $$\R^2$$ or $$\R^3$$) at which all the elements of its coordinates but $$x$$ are zero.

Y-Axis
In a cartesian coordinate system, the y-axis is the one usually depicted and visualised as going from "bottom" to "top" of the paper (or screen).

It consists of all the points in the Real Vector Space in question (usually either $$\R^2$$ or $$\R^3$$) at which all the elements of its coordinates but $$y$$ are zero.

Z-Axis
In a cartesian coordinate system, the z-axis is the axis passing through $$x = 0, y = 0$$. which is perpendicular to both the x-axis and the y-axis.

It consists of all the points in the Real Vector Space in question (usually $$\R^3$$) at which all the elements of its coordinates but $$z$$ are zero.

As the visual field is effectively two-dimensional, it is not possible to depict a three-dimensional space on a visual presentation (paper, screen etc.) directly.

Therefore the representation of the third axis of such a cartesian coordinate system is necessarily a compromise.

However, if we consider the plane of the visual field as being a representation of the x-y plane the z-axis can be imagined as coming "out of the page".

The usual convention for the orientation of the z-axis is that of the right-hand rule.

That is, if $$x$$ increases from "left to right", and $$y$$ increases from "bottom to top", then $$z$$ increases from "back to front" (that is, from "further away to closer in").