13. A homothety is defined by its center $S$ and coefficient $k$. Write the formula connecting $\vec{A}$ and $\vec{A}^{\prime}$, where $A^{\prime}$ is the image of point $A$ under this homothety. (Note that the answer does not depend on the choice of origin O.)
13. If a homothety with center $S$ and coefficient $k$ maps point $A$ to $A^{\prime}$, then $\left(\vec{A}^{\prime}-\vec{S}\right)=k(\vec{A}-\vec{S})$, from which:

$$
\vec{A}^{\prime}=\overrightarrow{k A}+(1-k) \vec{S}
$$