Is there an advantage over just essentially representing the 'area' as a
vector result perpendicular to the surface and giving it a size of width x
length? (in effect, using a cross-product vector approach.)

I have no idea of whether there is (was, more aptly) any advantage in
using one method over another in the calculations for box-making. But
the guy who gave the lecture stated that for him, not having to deal
with sign-changes shortened the time it took to make the calculation
and set up the machines. Using " i " allowed him to ignore sign
changes... I am sure today that all of this is done by computer, and
the set-up will be largely automated. Further, that the guys on the
floor very likely do not understand how those settings were derived.
Back then, at least some of them did.

Peter Wieck
Wyncote, PA