A regular hexagon has vertices ABCDEF and it is centered at the origin O. Express the following vectors in terms of a=OA and b=OBAB,BC,AD,CF,AC, and BC+DE+FA

Generally two laws of vectors are useed to  solve the problem: 1.triangle law 

2. parallogram law as shown if in fig1 and respectively.

In last fig hexgon drawn follow as per rules and  get solution like as............

from triangle OAB

AB=b-a

AD=OD-OA     (OD=-OA=-a)

AD=-2a

From triangle OBC

OB+BC=OC

To find OC   from triangle OAC...........OA+AC=OC.............          ( To find AC ..from OABC parllogram AC=-OA+AB=b-2a)

Now OC=a+b-2a=b-a

BC=-2a

remainig U can find similar procedure .

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Thanks a lot. but I don't see why BC would be -2a and not just -a