Question

ABCDEF is a regular hexagon. Ois the centre then prove that AB vector+ac vector+ ad vector +ae vector+ af vector=6AO vector

Answer 1

let us assume the hexagon be in I quadrant and one side( containing C) parallel to x-axis ( or i-axis). 
OG is vector = OC + CG. 
OC is vector = OA + CA. 
OA is vector =OB + BA 
BA , CA, CG have magnitudes predetermined by sides and angles predetermined by orientation of hexagon. 
then use theorum on vector transformations to generalise the result.