Orthocenter of an equilateral triangle ABC is the origin O. If `vec(OA)=veca, vec(OB)=vecb, vec(OC)=vecc`, then `vec(AB)+2vec(BC)+3vec(CA)=`

Options

A. `3vecc`
B. `3veca`
C. `vec0`
D. `3vecb`

Answer

B

Explanation

For an equilateral triangle, centroid is the same as orthocenter
`therefore (vec(OA) + vec(OB) + vec(OC))/(3) =vec0`
`therefore vec(OA)+vec(OB)+vec(OC) =vec0`
Now, `vec(AB) + 2vec(BC) + 3vec(CA)`
`=vec(OB) - vec(OA) +2vec(OC) - 2vec(OB) + 3vec(OA)-3vec(OC)`
`=-vec(OB) + 2vec(OA) -vec(OC)`
`=-(vec(OB) +vec(OA) +vec(OC))+3vec(OA)`
`=3vec(OA)`
`=3veca`