Definition:Cartesian 3-Space/Ordered Triple
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": coördinate.
But this is considered archaic nowadays and few sources still use it.
Sources
- 1934: D.M.Y. Sommerville: Analytical Geometry of Three Dimensions ... (previous) ... (next): Chapter $\text I$: Cartesian Coordinate-system: $1.1$. Cartesian coordinates
- 1936: Richard Courant: Differential and Integral Calculus: Volume $\text { II }$ ... (previous) ... (next): Chapter $\text I$: Preliminary Remarks on Analytical Geometry and Vector Analysis: $1$. Rectangular Co-ordinates and Vectors: $1$. Coordinate Axes
- 1947: William H. McCrea: Analytical Geometry of Three Dimensions (2nd ed.) ... (previous) ... (next): Chapter $\text {I}$: Coordinate System: Directions: $2$. Cartesian Coordinates
- 1972: Murray R. Spiegel and R.W. Boxer: Theory and Problems of Statistics (SI ed.) ... (previous) ... (next): Chapter $1$: Rectangular co-ordinates
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Cartesian coordinate system
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cartesian coordinate system