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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 and so on) 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".

Right-Hand Rule

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

Let the coordinate axes be oriented as follows:

Let the $x$-axis increase from West to East.
Let the $y$-axis increase from South to North.

Then the $z$-axis increases from below to above.

If the $x$-axis and $y$-axis are aligned with a piece of paper or a screen aligned perpendicular to the line of sight, this translates into the following orientation:

Let the $x$-axis increase from left to right.
Let the $y$-axis increase from bottom to top.

Then the $z$-axis increases from behind to in front (that is, from further away to closer in).

Also see

Linguistic Note

The plural of axis is axes, which is pronounced ax-eez not ax-iz.

Compare basis.