Definition:Cartesian 3-Space/Definition by Planes

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Definition

X-y-z-planes.png

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

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