Converting vectors into coordinates

Question

I was trying to solve a problem that involves vectors. So if we have three coordinates:
$A(p_1,q_1), B(p_2,q_2), C(p_3,q_3)$.
If we let: $\vec{AB} = \vec{a}, \vec{AC}=\vec{b}, \vec{BC}=\vec{b} - \vec{a}$
How do we convert the vectors back into coordinate if we have a position vector such that$\vec{AO}=\frac{1}{3} \vec{a} + \frac{1}{3}\vec{b} $
Thank you so much for you guys replies

Answer 1

From $\vec{A}=\vec{OA}=-\vec{AO}=-\frac13\vec{a}-\frac13\vec{b}$ we deduce $\vec{B}=\vec{OB}=\vec{OA}+\vec{AB}=\frac23\vec{a}-\frac13\vec{b}$, and similarly $\vec{C}=-\frac13\vec{a}+\frac23\vec{b}$.

Answer 2

You have that $\vec{a}=(p_2-p_1,q_2-q_1)$ and $\vec{b}=(p_3-p_1,q_3-q_1).$

You should now be able to find $$\frac13(\vec{a}+\vec{b})$$ in terms of the given coordinates $(p_i,q_i),$ where $i=1,2,3.$