Let G be the centroid of ΔABC. If $\vec{AB} = \vec a, \vec{AC} = \vec b$, then $\vec{AG}$, in terms of $\vec a$ and $\vec b$, is

$\frac{1}{3}(\vec a+\vec b)$

Let A be the origin. Then, $\vec{AB} = \vec a, \vec{AC} = \vec b$ implies that the position vectors of B and C are $\vec b$ and $\vec c$ respectively. Let AD be the median and G be the centroid. Then,

P.V. of $D+\frac{\vec a+\vec b}{2}$, P.V. of $G=\frac{\vec a+\vec b}{3}$

$∴\vec{AG}=\frac{\vec a+\vec b}{3}$