On an open ground , a motorist follows a track that turns to his left by an angle of ` 60^@` after every ` 500 m`. Starting from a given turn , specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with total path length covered by the motorist in each case.

Explanation

The path followed by him is a regular closed hexagon.
At third turn : The motor cyclist takes the third turn at C.
`therefore` Displacement `=|OvecC|=500+500=1000m`
Total path length from O to `C=OA+AB+BC`
`=500+500+500`
`=1500m`
At sixth turn: the cyclist takes the sixth turn at O .
Therefore total path length
`=OA+AB+BC+CD+DE+EO`
`=6xx500=3000m`
Displacement =`0`, As the cyclist come back to its initial position .
At eighth turn : The cyclist takes the eight turn at B .
Path length `=8xx500=4000m`
`therefore` Displacement =`|vecOB|`
`=sqrt((500)^(2)+(500)^(2)+2xx500xx500 cos 60^(@))`
`=866`
`=0.866 km`