# Definition:Cartesian 3-Space/Definition by Planes

## Contents

## Definition

Every point in ordinary $3$-space can be identified uniquely by means of an ordered triple of real coordinates $\tuple {x, y, z}$, as follows:

Identify one distinct point in space as the origin $O$.

Let $3$ distinct planes be constructed through $O$ such that all are perpendicular.

Each pair of these $3$ planes intersect in a straight line that passes through $O$.

Let $X$, $Y$ and $Z$ be points, other than $O$, one on each of these $3$ lines of intersection.

Then the lines $OX$, $OY$ and $OZ$ are named the $x$-axis, $y$-axis and $z$-axis respectively.

Select a point $P$ on the $x$-axis different from $O$.

Let $P$ be identified with the coordinate pair $\tuple {1, 0}$ in the $x$-$y$ plane.

Identify the point $P'$ on the $y$-axis such that $OP' = OP$.

Identify the point $P''$ on the $z$-axis such that $OP'' = OP$.

The orientation of the $z$-axis is determined by the position of $P''$ relative to $O$.

It is conventional to locate $P''$ as follows.

Imagine being positioned, standing on the $x$-$y$ plane at $O$, and facing along the $x$-axis towards $P$, with $P'$ on the left.

Then $P''$ is then one unit *above* the $x$-$y$ plane.

Let the $x$-$y$ plane be identified with the plane of the page or screen.

The orientation of the $z$-axis is then:

- coming vertically "out of" the page or screen from the origin, the numbers on the $z$-axis are positive
- going vertically "into" the page or screen from the origin, the numbers on the $z$-axis are negative.

### Coordinate Planes

Consider the **Cartesian $3$-space** defined by $3$ distinct perpendicular planes through the origin $O$.

These $3$ planes are known as the **coordinate planes** of the Cartesian $3$-space.

### Cartesian Coordinate 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 $\tuple {x, y, z}$.

## Also defined as

Some sources do not specify that the $3$ constructed distinct planes need to be perpendicular.

If they are not, then what results is an oblique coordinate system.

## Sources

- 1934: D.M.Y. Sommerville:
*Analytical Geometry of Three Dimensions*... (previous) ... (next): Chapter $\text I$: Cartesian Coordinate-system: $1.1$. Cartesian coordinates - 1947: William H. McCrea:
*Analytical Geometry of Three Dimensions*(2nd ed.) ... (previous) ... (next): Chapter $\text {I}$: Coordinate System: Directions: $2$. Cartesian Coordinates