If ABCDEF is regular hexagon, then AD+EB+FC is

Options

A. ` 2 A vec B `
B. `vec 0`
C. `3 A vec B `
D. `4 A vec B `

Answer

D

Explanation

Let ` A vec B = vec a and B vec C = vec b .` Then ,
`A vec O = B vec C = vec b , Ovec C = F vec O = vec a `

In ` Delta AOB`, we have
`A vec B + B vec O = A vec O `
`rArr vec a + B vec O = vec b " " [ because A vec O = B vec C =vec b]`
` rArr B vec O = vec b - vec a `
`rArr B vec E = 2 B vec O = 2 (vec b - vec a )`
`therefore A vec D + E vec B + F vec C = A vec D - B vec E + 2 O vec C `
`rArr A vec D + E vec B + F vec C = 2 vec b -2 (vec b - vec a ) + 2 vec a =4 vec a = 4 A vec B `