Let the side of the hexagon be length r, which is also the radius of the circle on which it is inscribed. The 6 vertices can be represented by the following coordinates:
A r(1,0); B r(1/2,sqrt(3)/2); C r(-1/2,sqrt(3)/2); D r(-1,0); E r(-1/2,-sqrt(3)/2); F r(1/2, -sqrt(3)/2).
The unit vectors for x and y are i and j, so the vectors are A=ri and B=(r/2)i+(rsqrt(3)/2)j. So it's clear that D=-A=-a and E=-B=-b. A-B=a-b=(r/2)i-(rsqrt(3)/2)j=F and C=-F=b-a.
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