If `O\ a n d\ O^(prime)` are circumcentre and orthocentre of `\ A B C ,\ t h e n\ vec O A+ vec O B+ vec O C` equals `2 vec O O '` b. ` vec O O '` c. ` vec O ' O` d. `2 vec O ' O`

Options

A. `vec(O'O)`
B. `vec(O O')`
C. `2vec(O O')`
D. `vec(0)`

Answer

B

Explanation

Let G be the centroid of `Delta ABC.` Then O', G, O are collinear and G divides O' O in the ratio `2:1 " i.e. " (O'G)/(OG)=(2)/(1)`
Replacing S by O in illustration 2, we have
`vec(OA)+vec(OB) + vec(OC)=3O vec(G)`
`rArr vec(OA) + vec(OB)+vec(OC)=2vec(OG)+vec(OG)`
`rArr O vec(A) + Ovec(B) + O vec(C) = G vec(O')+Ovec(G) " "[ because 2OG=GO']`
`rArr O vec(A) +O vec(B) + O vec(C) = vec(O O')`