5.12. Let $O$ be the point of intersection of the medians of triangle $ABC$ and $\overline{A O}=\bar{a}, \overline{A C}=\bar{b}$. Decompose $\overline{A B}$ and $\overline{B C}$ in terms of the vectors $\bar{a}$ and $\bar{b}$.
5.12. Since $O$ is the point of intersection of the medians of $\triangle ABC$ (Fig. 5.7), then $\overline{AO}=\frac{2}{3} \overline{AM}_{1}$; thus $\overline{AM}_{1}=\frac{3}{2} \bar{a}, \overline{M_{1}C}=\overline{AC}-\overline{AM}_{1}=\bar{b}-\frac{3}{2} \bar{a}$, $BC=2 \overline{M_{1}C}=2 \bar{b}-3 \bar{a}, \overline{AB}=\overline{AM}_{1}+\overline{M_{1}B}=\overline{AM}_{1}-\overline{M_{1}C}=\frac{3}{2} \bar{a}-$ $-\left(\bar{b}-\frac{3}{2} \bar{a}\right)=3 \bar{a}-\bar{b}$

Answer: $\overline{AB}=3 \bar{a}-\bar{b}, \overline{BC}=2 \bar{b}-3 \bar{a}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-603.jpg?height=313&width=615&top_left_y=609&top_left_x=277)

Fig. 5.7