If vectors AB, AC, AD, AE, and AF act at the regular hexagon ABCDEF. Prove that the resultant of these vectors is 6 AO where O is the center of the hexagon.

Resultant = AB + AC + AD + AE+ AF

= AB + (AC + AF) + AD+ AE

Also CD = AF ⇒ AC +CD = AD

Resultant = AB + AD + AD + AE

= 2AD + AB + AE

Also, DE = AB

Resultant = 2AD + DE + AE

= 2AD + AD

= 3AD

Since D is exact opposite to A

⇒ AO + OD = AD where O is mid point

AO = OD

⇒ AD = 2AO

3AD = 6AO

So, Resultant = 6AO