Definition:Cartesian 3-Space/Orientation

Definition

Let the $x$-axis, $y$-axis and $z$-axis be defined.

Let a point $P$ be identified on the $x$-axis, different from $O$, with the coordinate pair $\tuple {1, 0}$ in the $x$-$y$ plane.

Let the point $P'$ be identified on the $y$-axis such that $OP' = OP$.

It remains to identify the point $P$ on the $z$-axis such that $OP = OP$.

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.

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.