# Definition:Cartesian Product/Cartesian Space/Real Cartesian Space

## Definition

Let $n \in \N_{>0}$.

Then $\R^n$ is the cartesian product defined as follows:

- $\ds \R^n = \underbrace {\R \times \R \times \cdots \times \R}_{\text {$n$ times} } = \prod_{k \mathop = 1}^n \R$

Similarly, $\R^n$ can be defined as the set of all real $n$-tuples:

- $\R^n = \set {\tuple {x_1, x_2, \ldots, x_n}: x_1, x_2, \ldots, x_n \in \R}$

### Cartesian Plane

The **Cartesian plane** is a Cartesian coordinate system of $2$ dimensions.

Every point on the plane can be identified uniquely by means of an ordered pair of real coordinates $\tuple {x, y}$, as follows:

Identify one distinct point on the plane as the origin $O$.

Select a point $P$ on the plane different from $O$.

Construct an infinite straight line through $O$ and $P$ and call it the $x$-axis.

Identify the $x$-axis with the real number line such that:

The orientation of the $x$-axis is determined by the relative positions of $O$ and $P$.

It is conventional to locate $P$ to the right of $O$, so as to arrange that:

- to the right of the origin, the numbers on the $x$-axis are positive
- to the left of the origin, the numbers on the $x$-axis are negative.

Construct an infinite straight line through $O$ perpendicular to the $x$-axis and call it the $y$-axis.

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

Identify the $y$-axis with the real number line such that:

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

It is conventional to locate $P'$ such that, if one were to imagine being positioned at $O$ and facing along the $x$-axis towards $P$, then $P'$ is on the left.

Hence with the conventional orientation of the $x$-axis as horizontal and increasing to the right:

- going vertically "up" the page or screen from the origin, the numbers on the $y$-axis are positive
- going vertically "down" the page or screen from the origin, the numbers on the $y$-axis are negative.

### Cartesian 3-Space

The **Cartesian $3$-space** is a Cartesian coordinate system of $3$ dimensions.

### Definition by Axes

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:

Construct a Cartesian plane, with origin $O$ and axes identified as the $x$-axis and $y$-axis.

Recall the identification of the point $P$ with the coordinate pair $\tuple {1, 0}$ in the $x$-$y$ plane.

Construct an infinite straight line through $O$ perpendicular to both the $x$-axis and the$y$-axis and call it the $z$-axis.

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

Identify the $z$-axis with the real number line such that:

### Definition by Planes

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 $OP' ' = OP$.

### Orientation

It remains to identify the point $P' '$ on the $z$-axis such that $OP' ' = OP$.

#### Right-Handed

The Cartesian $3$-Space is defined as **right-handed** when $P' '$ is located as follows.

Let the coordinate axes be oriented 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.

#### Left-Handed

The Cartesian $3$-Space is defined as **left-handed** when $P' '$ is located as follows.

Let the coordinate axes be oriented 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.

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

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

### Countable-Dimensional Real Cartesian Space

The countable cartesian product defined as:

- $\ds \R^\omega := \R \times \R \times \cdots = \prod_\N \R$

is called the **countable-dimensional real cartesian space**.

Thus, $\R^\omega$ can be defined as the set of all real sequences:

- $\R^\omega = \set {\sequence {x_1, x_2, \ldots}: x_1, x_2, \ldots \in \R}$

## Also known as

The **real cartesian space** of order $n$ is sometimes seen as the **(real) cartesian $n$-space**.

Some sources call this **euclidean $n$-space** -- however, on $\mathsf{Pr} \infty \mathsf{fWiki}$ this term is reserved for the associated metric space.

## Also see

It can be shown that:

- $\R^2$ is isomorphic to any infinite flat plane in space
- $\R^3$ is isomorphic to the whole of space itself.

## Source of Name

This entry was named for RenĂ© Descartes.

## Sources

- 1963: Louis Auslander and Robert E. MacKenzie:
*Introduction to Differentiable Manifolds*... (next): Euclidean, Affine, and Differentiable Structure on $R^n$ - 1965: Claude Berge and A. Ghouila-Houri:
*Programming, Games and Transportation Networks*... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.1$. Sets - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): Notation for Some Important Sets - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 5$: Products of Sets - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 5$: Cartesian Products - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Example $2.1.4$