If G is the centroid of a triangle ABC, prove that `vec(GA)+vec(GB)+vec(GC)=vec(0)`.

Options

A. `vec0`
B. `3 vec(GA)`
C. `3 vec(GB)`
D. `3 vec(GC)`

Answer

A

Explanation

We have `vec(GB) + vec(GC) = (1+1) vec(GD) = 2vec(GD)`, where D is the midpoint of BC. Therefore,
`vec(GA) = vec(GB) +vec(GC) = vec(GA)+ 2vec(GD) `
`" "= vec(GA) - vec(GA) =0`
`(because G` divides AC in the ratio `2:1, therefore 2 vec(GD)=- vec(GA))`