3.72: Vectors in a triangle

The diagram shows a triangle $OAC$. The point $B$ lies on $AC$ such that $AB:AC = 2:5$. It is given that $\overrightarrow{OA}=\mathbf{a}$, $\overrightarrow{OB}=\mathbf{b}$ and $\overrightarrow{OC}=\mathbf{c}$.

Find expressions for $\overrightarrow{AB}$ and $\overrightarrow{BC}$ in terms of $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$, and hence show the required vector relation in the mark scheme.