3.47: Vector geometry in triangle $OAB$

The diagram shows triangle $OAB$ with $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$. Point $X$ lies on $OA$ with $\overrightarrow{OX}=\tfrac{3}{5}\mathbf{a}$. Point $Y$ is the mid-point of $AB$.

Find $\overrightarrow{AB}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

Find $\overrightarrow{XY}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

The lines $OB$ and $XY$ extended meet at $Z$. Given $\overrightarrow{YZ}=m\overrightarrow{XY}$ and $\overrightarrow{BZ}=n\mathbf{b}$, find $\overrightarrow{XZ}$ in terms of $m$, $\mathbf{a}$ and $\mathbf{b}$.

Find $\overrightarrow{XZ}$ in terms of $n$, $\mathbf{a}$ and $\mathbf{b}$.

Hence find the values of $m$ and $n$.