Definition:Cartesian Plane/Ordered Pair

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Identification of Point in Plane with Ordered 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 $\tuple {x, y}$.


$x$ 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.


Also known as

The ordered pair $\tuple {x, y}$ which determine the location of $P$ in the cartesian plane can be referred to as the rectangular coordinates or (commonly) just coordinates of $P$.


Linguistic Note

It's an awkward word coordinate.

It really needs a hyphen in it to emphasise its pronunciation (loosely and commonly: coe-wordinate), and indeed, some authors spell it co-ordinate.

However, this makes it look unwieldy.


An older spelling puts a diaeresis indication symbol on the second "o": coördinate.

But this is considered archaic nowadays and few sources still use it.


Sources

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