Definition:Cartesian 3-Space/Definition by Planes
Definition
Every point in ordinary $3$-space can be identified uniquely by means of an ordered triple of real coordinates $\tuple {x, y, z}$, as follows:
Identify one distinct point in space as the origin $O$.
Let $3$ distinct planes be constructed through $O$ such that all are perpendicular.
Each pair of these $3$ planes intersect in a straight line that passes through $O$.
Let $X$, $Y$ and $Z$ be points, other than $O$, one on each of these $3$ lines of intersection.
Then the lines $OX$, $OY$ and $OZ$ are named the $x$-axis, $y$-axis and $z$-axis respectively.
Select a point $P$ on the $x$-axis different from $O$.
Let $P$ be identified with the coordinate pair $\tuple {1, 0}$ in the $x$-$y$ plane.
Identify the point $P'$ on the $y$-axis such that $OP' = OP$.
Identify the point $P$ on the $z$-axis such that $OP = OP$.
Orientation
It remains to identify the point $P$ on the $z$-axis such that $OP = OP$.
Right-Handed
The Cartesian $3$-Space is defined as right-handed when $P$ is located as follows.
Let the coordinate axes be oriented as follows:
Imagine being positioned, standing on the $x$-$y$ plane at $O$, and facing along the $x$-axis towards $P$, with $P'$ on the left.
Then $P$ is then one unit above the $x$-$y$ plane.
Left-Handed
The Cartesian $3$-Space is defined as left-handed when $P$ is located as follows.
Let the coordinate axes be oriented as follows:
Imagine being positioned, standing on the $x$-$y$ plane at $O$, and facing along the $x$-axis towards $P$, with $P'$ on the left.
Then $P$ is then one unit below the $x$-$y$ plane.
Coordinate Planes
Consider the Cartesian $3$-space defined by $3$ distinct perpendicular planes through the origin $O$.
These $3$ planes are known as the coordinate planes of the Cartesian $3$-space.
Cartesian Coordinate 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 $\tuple {x, y, z}$.
Also defined as
Some sources do not specify that the $3$ constructed distinct planes need to be perpendicular.
If they are not, then what results is an oblique coordinate system.
Sources
- 1934: D.M.Y. Sommerville: Analytical Geometry of Three Dimensions ... (previous) ... (next): Chapter $\text I$: Cartesian Coordinate-system: $1.1$. Cartesian coordinates
- 1947: William H. McCrea: Analytical Geometry of Three Dimensions (2nd ed.) ... (previous) ... (next): Chapter $\text {I}$: Coordinate System: Directions: $2$. Cartesian Coordinates
- 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.) ... (next): Chapter $2$ Coordinate Systems $2.1$ Curvilinear Coordinates