Maximum of Three Mutually Perpendicular Lines in Ordinary Space

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In ordinary space, there can be no more than $3$ straight lines which are pairwise perpendicular.

Thus, in a configuration of $4$ straight lines in space, at least one pair will not be perpendicular to each other.


By assumption and popular belief, ordinary space (on the measurable local level) is an instance of a $3$-dimensional Euclidean space.

Hence the results of Euclid's The Elements can be applied.

Let there be $4$ straight lines $a$, $b$, $c$ and $d$ in space.

Let $a$ be perpendicular to each of $b$, $c$ and $d$.

From Three Intersecting Lines Perpendicular to Another Line are in One Plane, $b$, $c$ and $d$ are all in the same plane.

Let $b$ be perpendicular to $c$ and $d$.

Either $c$ and $d$ are the same line or not.

If $c = d$ then $c$ is not perpendicular to $d$.

Otherwise, from Equal Corresponding Angles implies Parallel Lines, $c$ is parallel to $d$.

In either case $c$ is not perpendicular to $d$.

Hence the result.