Definition:Cartesian 3-Space/Ordered Triple

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


$x$ 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.


$z$ Coordinate

Let $z$ be the length of the line segment from the origin $O$ to the foot of the perpendicular from $Q$ to the $z$-axis.

Then $z$ is known as the $z$ coordinate.

If $Q$ is in the positive direction along the real number line that is the $z$-axis, then $z$ is positive.

If $Q$ is in the negative direction along the real number line that is the $z$-axis, then $z$ is negative.


Also known as

The ordered triple $\tuple {x, y, z}$ which determines the location of $P$ in the cartesian $3$-space 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": co├Ârdinate. But this is considered archaic nowadays and few sources still use it.


Sources