If $\overrightarrow{a}$ and $\overrightarrow{b}$ represent two adjacent sides $\overrightarrow{AB}$ and $\overrightarrow{BC}$ respectively of a parallelogram $ABCD$, find the diagonals $\overrightarrow{AC}$ and $\overrightarrow{BD}$.
$ABCD$ is a parallelogram.
$\therefore$ $\;$ $AD = BC$ and $AD \parallel BC$.
Given: $\overrightarrow{AB} = \overrightarrow{a}$; $\;$ $\overrightarrow{BC} = \overrightarrow{b}$
Diagonal $\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{a} + \overrightarrow{b}$
$\because$ $\;$ $AD = BC$ and $AD \parallel BC$,
$\therefore$ $\;$ $\overrightarrow{AD} = \overrightarrow{BC} = \overrightarrow{b}$
Now, $\overrightarrow{BA} = - \overrightarrow{AB} = - \overrightarrow{a}$
Diagonal $\overrightarrow{BD} = \overrightarrow{BA} + \overrightarrow{AD} = - \overrightarrow{a} + \overrightarrow{b}$