A point `O` is the centre of a circle circumscribed about a triangle`A B Cdot` Then ` vec O Asin2A+ vec O Bsin2B+ vec O Csin2C` is equal to a. `( vec O A""+ vec O B""+ vec O C)sin2A` b. `3 vec O G ,w h e r eG` is the centroid of triangle `A B C` c. ` vec0` d. none of these

Options

A. `(vec(OA) + vec(OB) + vec(OC)) sin 2A`
B. ` 3 vec(OG)`, where G is the centroid of triangle ABC
C. `vec0`
D. none of these

Answer

C

Explanation

The position vector of the point O with respect to itself is
`(vec(OA) sin 2A + vec(OB) sin2B + vec(OC) sin 2C)/(sin 2A + sin 2B + sin 2C)`
`rArr (vec(OA) sin 2A + vec(OB) sin 2B + vec(OC) sin 2C)/( sin2A + sin 2B + sin 2C) = vec0`
or `" "vec(OA) sin 2A + vec(OB) sin 2B + vec(OC) sin 2C = vec0`