Definition:Cartesian Product/Cartesian Space/Two Dimensions

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Definition

Let $S$ be a set.

The cartesian $2$nd power of $S$ is:

$\displaystyle S^2 = S \times S = \set {\tuple {x_1, x_2}: x_1, x_2 \in S}$


Thus $S^2 = S \times S$


The set $S^2$ called a cartesian space of $2$ dimensions.


Cartesian Plane

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


A generic point in the Cartesian Coordinate Plane

The points in the plane can be identified uniquely by means of a pair of coordinates.

Two perpendicular straight lines are chosen. These are understood to be infinite. These are called the axes.

The usual directions to make these are:

$(1): \quad$ Across the page, from left to right. This is usually called the $x$-axis.
$(2): \quad$ Up the page, from bottom to top. This is usually called the $y$-axis.

The point of intersection of the axes is called the origin.

A unit length is specified.

The axes are each identified with the set of real numbers $\R$, where the origin is identified with $0$.

The real numbers increase to the right on the $x$-axis, upwards on the $y$-axis.


Thus the plane can be identified with the cartesian product $\R^2$.

In this context, $\R^2$ is called the (cartesian) coordinate plane.


Identification of Point in Plane with Ordered Pair

Every point on the plane can be identified by means of a pair of 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 $O$.

Let the distance from the origin to $P$ be defined as being $1$.

Draw an infinite straight line through $O$ and $P$ and call it the $X$-axis.

Draw an infinite straight line through $O$ perpendicular to $OP$ and call it the $Y$-axis.


Now, let $Q$ be any point on the plane.

Draw two lines through $Q$, parallel to the $X$-axis and $Y$-axis.

The plane is then conventionally oriented so that the $X$-axis is horizontal with $P$ being to the right of $O$.


Thus the $Y$-axis is then a vertical line.


Thus the point $Q$ can be uniquely identified by the ordered pair $\tuple {x, y}$ as follows:


X Coordinate

The distance of the line segment from $Q$ to the $Y$-axis is known as the $X$ coordinate and (usually) denoted $x$.

If $Q$ is to the right of the $Y$-axis, then $x$ is positive.

If $Q$ is to the left of the $Y$-axis, then $x$ is negative.

The $X$ coordinate of all points on the $Y$-axis is zero.


Y Coordinate

The distance of the line segment from $Q$ to the $X$-axis is known as the $Y$ coordinate and (usually) denoted $y$.

If $Q$ is above the $X$-axis, then $y$ is positive.

If $Q$ is below the $X$-axis, then $y$ is negative.

The $Y$ coordinate of all points on the $X$-axis is zero.


The point $P$ is identified with the coordinates $\tuple {1, 0}$.


Source of Name

This entry was named for René Descartes.


Sources