In a regular hexagon `A B C D E F ,\ A vec B=a ,\ B vec C= vec b\ a n d\ vec C D=cdotT h e n\ vec A E=` ` vec a+ vec b+ vec c` b. `2 vec a+ vec b+ vec c` c. ` vec b+ vec c` d. ` vec a+2 vec b+2 vec c`

Options

A. `veca + vec b + vec c`
B. `2 veca + vec b + vec c `
C. `vec a + vec c `
D. `vec a + 2 vec b + 2 vec c `

Answer

C

Explanation

We have,
`A vec B = vec a , B vec C = vec b and C vec D = vec c `
`therefore A vec C = vec a + vec b `
In ` Delta ACD`, we have
`A vec C + C vec D = A vec D `
`rArr vec a + vec b + vec c = A vec D `
In ` Delta AED`, we have
`A vec D + D vec E = A vec E `
`rArr vec a + vec b + vec c - vec a = A vec E " " [ because D vec E = B vec A = - vec a ]`
` rArr A vec E = vec b + vec c `