Definition:Cartesian 3-Space/Orientation
Definition
Consider a Cartesian $3$-Space.
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 $OP = OP$.
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.