given a hexagon of sides AB,BC,CD,DE,EF,FA with centre at Othen prove that AB+AC+AD+AE+AF=6AO

If R be the resultant, then

R=AB+AC+AD+AE+AF

=AB+(AD+DC)+AD+(AD+DE)+AF

Now AF and DC are two equal vectors in opposite directions and hence they cancel each other.

Similarly AB and DE cancel each other.

so R = 3A

=6AO

where is the mid. point of AD i. e. the centroid of the hexagon.