Definition:Axis/Z-Axis/Right-Hand Rule

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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).