$+)\quad \overrightarrow{BI}=\overrightarrow{BA}+\overrightarrow{AI}$
$=\overrightarrow{BA} +\dfrac13\overrightarrow{AC}$
$=\overrightarrow{BA} +\dfrac13(\overrightarrow{AB} +\overrightarrow{BC})$
$=\overrightarrow{BA} -\dfrac13\overrightarrow{BA} +\dfrac13\overrightarrow{BC}$
$=\dfrac23\overrightarrow{BA} +\dfrac13\overrightarrow{BC}$
$+)\quad \overrightarrow{BK}=\dfrac{1}{2}(\overrightarrow{BA} +\overrightarrow{BN})$
$=\dfrac12\overrightarrow{BA} +\dfrac14\overrightarrow{BC}$