If S is the circumcentre, G the centroid, O the orthocentre of a triangle ABC, then $\vec{SA}+\vec{SB}+\vec{SC}$ is:

$3\vec{SG}$

We know G divides S (circumcentre) and O (orthocenter) in the ratio of 1 : 2

$⇒\vec{G}=\frac{2\vec{S}+\vec O}{2+1}=\frac{2\vec S}{3}$ … (i)

Now, $\vec{SA}+\vec{SB}+\vec{SC}=\vec a+\vec b+\vec c-3\vec s$

But $\vec G=\frac{\vec a+\vec b+\vec c}{3}$

$⇒\vec{SA}+\vec{SB}+\vec{SC}=3\vec G-3\vec S=3\vec{SG}$

Also $3\vec G-3\vec S=2\vec S-3\vec -\vec S=\vec{SO}$ …(Using (i))