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... F is a vector. That is sufficient (indeed more than sufficient) to
guarantee that we can write it as the sum of three other forces
F1, F2, and F3.
I hate to belabor the obvious, but the acceleration a is also a
vector. That means we can write it as the sum of three other
accelerations a1, a2, and a3.
We can start with the obvious possibility a1=F1/m, a2=F2/m, and a3=F3/m
... but there is an infinitude of other ways of writing a as the sum
of three vectors ... not to mention an additional infinitude of ways of
writing a as the sum of four vectors, et cetera. ...