Five forces and vector(AB), vector(AC), vector(AD), vector(AD), vector(AE) and vector(AF) act act at the vertex of a regular hexagon ABCDEF.

Question

Five forces and \(\overset{\longrightarrow}{AB},\overset{\longrightarrow}{AC},\overset{\longrightarrow}{AD},\overset{\longrightarrow}{AE} \)and \(\overset{\longrightarrow}{AF} \) act act at the vertex of a regular hexagon ABCDEF. Prove that the resultant is 6 \(\overset{\longrightarrow}{AO} \) where O is the centre of hexagon. 

Answer 1

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

Therefore the resultant of the five forces

Hence proved