# Definition:Cartesian 3-Space/Ordered Triple

## Contents

## Identification of Point in Space with Ordered Triple

Let $Q$ be a point in Cartesian $3$-space.

Construct $3$ straight lines through $Q$:

- one perpendicular to the $y$-$z$ plane, intersecting the $y$-$z$ plane at the point $x$

- one perpendicular to the $x$-$z$ plane, intersecting the $x$-$z$ plane at the point $y$

- one perpendicular to the $x$-$y$ plane, intersecting the $x$-$y$ plane at the point $z$.

The point $Q$ is then uniquely identified by the ordered pair $\tuple {x, y, z}$.

### $x$ Coordinate

Let $x$ be the length of the line segment from the origin $O$ to the foot of the perpendicular from $Q$ to the $x$-axis.

Then $x$ is known as the **$x$ coordinate**.

If $Q$ is in the positive direction along the real number line that is the $x$-axis, then $x$ is positive.

If $Q$ is in the negative direction along the real number line that is the $x$-axis, then $x$ is negative.

### $y$ Coordinate

Let $y$ be the length of the line segment from the origin $O$ to the foot of the perpendicular from $Q$ to the $y$-axis.

Then $y$ is known as the **$y$ coordinate**.

If $Q$ is in the positive direction along the real number line that is the $y$-axis, then $y$ is positive.

If $Q$ is in the negative direction along the real number line that is the $y$-axis, then $y$ is negative.

### $z$ Coordinate

Let $z$ be the length of the line segment from the origin $O$ to the foot of the perpendicular from $Q$ to the $z$-axis.

Then $z$ is known as the **$z$ coordinate**.

If $Q$ is in the positive direction along the real number line that is the $z$-axis, then $z$ is positive.

If $Q$ is in the negative direction along the real number line that is the $z$-axis, then $z$ is negative.

## Also known as

The ordered triple $\tuple {x, y, z}$ which determines the location of $P$ in the cartesian $3$-space can be referred to as the **rectangular coordinates** or (commonly) just **coordinates** of $P$.

## Linguistic Note

It's an awkward word **coordinate**. It really needs a hyphen in it to emphasise its pronunciation (loosely and commonly: **coe-wordinate**), and indeed, some authors spell it **co-ordinate**. However, this makes it look unwieldy.

An older spelling puts a diaeresis indication symbol on the second "o": **coördinate**. But this is considered archaic nowadays and few sources still use it.

## Sources

- 1934: D.M.Y. Sommerville:
*Analytical Geometry of Three Dimensions*... (previous) ... (next): Chapter $\text I$: Cartesian Coordinate-system: $1.1$. Cartesian coordinates - 1936: Richard Courant:
*Differential and Integral Calculus: Volume $\text { II }$*... (previous) ... (next): Chapter $\text I$: Preliminary Remarks on Analytical Geometry and Vector Analysis: $1$. Rectangular Co-ordinates and Vectors: $1$. Coordinate Axes - 1947: William H. McCrea:
*Analytical Geometry of Three Dimensions*(2nd ed.) ... (previous) ... (next): Chapter $\text {I}$: Coordinate System: Directions: $2$. Cartesian Coordinates - 1972: Murray R. Spiegel and R.W. Boxer:
*Theory and Problems of Statistics*(SI ed.) ... (previous) ... (next): Chapter $1$: Rectangular co-ordinates - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Cartesian coordinate system** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Cartesian coordinate system**