Gọi $G$ là trọng tâm $\triangle ABC$
$\Rightarrow \begin{cases}\overrightarrow{BG} = \dfrac23\overrightarrow{BM}\\\overrightarrow{GK} = \dfrac13\overrightarrow{AK}\end{cases}$
Ta được:
$\bullet\quad \overrightarrow{BC} = 2\overrightarrow{BK}$
$\Leftrightarrow \overrightarrow{BC} = 2\left(\overrightarrow{BG} + \overrightarrow{GK}\right)$
$\Leftrightarrow \overrightarrow{BC} = 2\left(\dfrac23\overrightarrow{BM} + \dfrac13\overrightarrow{AK}\right)$
$\Leftrightarrow \overrightarrow{BC} = \dfrac23\overrightarrow{AK} + \dfrac43\overrightarrow{BM}$
$\bullet\quad \overrightarrow{CA} = \overrightarrow{CK} + \overrightarrow{KA}$
$\Leftrightarrow \overrightarrow{CA} = -\dfrac12\overrightarrow{BC} -\overrightarrow{AK}$
$\Leftrightarrow \overrightarrow{CA} = -\dfrac12\left(\dfrac23\overrightarrow{AK} + \dfrac43\overrightarrow{BM}\right) - \overrightarrow{AK}$
$\Leftrightarrow \overrightarrow{CA} =-\dfrac43\overrightarrow{AK} - \dfrac23\overrightarrow{BM}$
