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Without thinking about it more deeply, I have assumed the the torque/work
case implies that there is a dimensional difference between "parallel
metres" and "perpendicular metres", i.e. between the result of a dot product
and of a cross product. Can quantities have the same dimensions and yet be
completely different physically?
Mark
Yes. The present example is a good one. For a related example which
will give you some reason to think about such things I recommend
reading the first chapter in "Spacetime Physics" by Edwin F. Taylor
and Jonathan Archibald Wheeler. It is called "The Parable of the
Surveyors". The fact that the dimensions of torque and energy are
the same is an artifact only of the conventional manner in which
these quantities were separately introduced into physics.
Leigh