# Definition:Axis/Z-Axis

## Definition

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".

## 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:

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:

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.

## Sources

- 1967: D.E. Bourne and P.C. Kendall:
*Vector Analysis*... (previous) ... (next): Chapter $1$: Rectangular Cartesian Coordinates and Rotation of Axes: $1.1$ Rectangular cartesian coordinates - 1972: Murray R. Spiegel and R.W. Boxer:
*Theory and Problems of Statistics*(SI ed.) ... (previous) ... (next): Chapter $1$: Rectangular co-ordinates - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**$z$-axis** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**$z$-axis** - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $6$: Curves and Coordinates: Descartes - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**$z$-axis**