Five forces and and act at the vertex of a regular hexagon ABCDEF. Prove that the resultant is 6 where O is the centre of hexagon.

Question

Five forces and  and  act at the vertex of a regular hexagon ABCDEF. Prove that the resultant is 6  where O is the centre of hexagon.

Philoid · Accepted Answer

Given a regular hexagon ABCDEF with O as the centre of the hexagon as shown in figure below

To prove

We know that centre O of the hexagon bisects the diagonals

∴, ,

Consider ΔABO and apply triangle law of vector, we get

And consider ΔACO and apply triangle law of vector, we get

And consider ΔAEO and apply triangle law of vector, we get

And consider ΔAFO and apply triangle law of vector, we get

Now,

Substitute the corresponding values from eqn(i) to eqn(v) in above eqn, we get

[from eqn(i)]

[as and ]

Hence

Therefore the resultant of the five forces and is

Hence proved