Definition:Cartesian 3-Space/Orientation/Right-Handed
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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 $OP = OP$.
The orientation of the $z$-axis is determined by the position of $P$ relative to $O$.
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.
Hence, let the coordinate axes be oriented as follows:
Then the $z$-axis increases from below to above.
Simiarly, let the $x$-$y$ plane be identified with the plane of the page or screen such aligned perpendicular to the line of sight such that:
Then the $z$-axis increases from behind to in front (that is, from further away to closer in).