Definition:Cartesian Plane

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Definition

A general point $Q = \tuple {x, y}$ in the 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:

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

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:

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


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 Coordinate Pair

Let $Q$ be a point on the Cartesian plane.

Construct two straight lines through $Q$:

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

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


Hence:

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


$x$ 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.


Quadrants

For ease of reference, the cartesian plane is often divided into four quadrants by the axes:


First Quadrant

Quadrant-I.png

Quadrant $\text{I}: \quad$ The area above the $x$-axis and to the right of the $y$-axis is called the first quadrant.

That is, the first quadrant is where both the $x$ coordinate and the $y$ coordinate of a point are positive.


Second Quadrant

Quadrant-II.png

Quadrant $\text{II}: \quad$ The area above the $x$-axis and to the left of the $y$-axis is called the second quadrant.

That is, the second quadrant is where the $x$ coordinate of a point is negative and the $y$ coordinate of a point is positive.


Third Quadrant

Quadrant-III.png

Quadrant $\text{III}: \quad$ The area below the $x$-axis and to the left of the $y$-axis is called the third quadrant.

That is, the third quadrant is where both the $x$ coordinate and the $y$ coordinate of a point are negative.


Fourth Quadrant

Quadrant-IV.png

Quadrant $\text{IV}: \quad$ The area below the $x$-axis and to the right of the $y$-axis is called the fourth quadrant.

That is, the fourth quadrant is where the $x$ coordinate of a point is positive and the $y$ coordinate of a point is negative.


Note that the axes themselves are generally not considered to belong to any quadrant.


Also known as

The cartesian coordinate plane is often seen referred to as the $x y$-plane, or (without the hyphen) the $x y$ plane.

Some sources refer to it as the Euclidean plane, but on $\mathsf{Pr} \infty \mathsf{fWiki}$ that term is reserved for the abstract geometry consisting of $\R^2$ together with the set of straight lines.


Historical Note

The Cartesian plane was supposedly invented by René Descartes in around the year $1637$.

However, a study of the literature will reveal that this idea originated considerably earlier, perhaps going back as far as Nicole Oresme.


Sources

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