Definition:Cartesian Product/Cartesian Space/Three Dimensions

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Definition

Let $S$ be a set.

The cartesian $3$rd power of $S$ is:

$S^3 = S \times S \times S = \set {\tuple {x_1, x_2, x_3}: x_1, x_2, x_3 \in S}$


The set $S^3$ called a cartesian space of $3$ dimensions.


Cartesian 3-Space

When $S$ is the set of real numbers $\R$, the cartesian product takes on a special significance.


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


Definition by Axes

A general point in Cartesian $3$-Space

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:

$0$ is identified with the origin $O$
$1$ is identified with the point $P$


Definition by Planes

X-y-z-planes.png

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.


Also see


Source of Name

This entry was named for René Descartes.