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The question is not clearly mentioned.  Is it that like the following ?

vector(AG)=\vec{a}\\vector(BG)=\vec{b}\\vector(CG)=\vec{c}\\\\vector(AB)=vector(AG)+vector(GB)\\.\ \ \ \ =\ vector(AG)-vector(BG)\\\\\vec{AB}=\vec{a}-\vec{b}
if the question is like:

let\ vector(AB)=\vec{a}\ \ and\ \ vector(AC)=\vec{b}\\then,\ \ vector(BC)=vector(BA)+vector(AC)\\=-vector(AB)+vector(AC)=\vec{b}-\vec{a}

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If g be a centroid of triangle abc such that vector(ab)=vector(a),vector(ac)=vector(b) then vector(ab)=?
where is the need of the centroid ?
if vector (AG) = vec{a} , vector (BG) = vec{b} then, vector(CG) = - vec{a} - vec{b}. this is if we are talking about the lines joining vertices and the centroid.