Five forces ` vec A B` , ` vec A C` , ` vec A D` , ` vec A E` and ` vec A F` act at the vertex of a regular hexagon `A B C D E Fdot` Prove that the resultant is `6 vec A O` , where `O` is the centre of hexagon.

Explanation

`vec(AB)+vec(BO)=vec(AO)`

`vec(AC)+vec(CO)=vec(AO)`

`vec(AD)+vec(DO)=vec(AO)`

`vec(AE)+vec(EO)=vec(AD)`

`vec(AF)+vec(FO)=vec(AO)`

`(vec(AB)+vec(AC)+vec(AD)+vec(AB)+vec(AF))+(vec(BO)+vec(CO)+vec(DO)+vec(EO)+vec(FO))=5vec(AO)`

`x+vec(AO)=5vec(AO)`

`x=6vec(AO)`.