Question

ABCD is a quadrilateral. Find the sum the vectors \[\overrightarrow{BA} , \overrightarrow{BC} , \overrightarrow{CD}\] and \[\overrightarrow{DA}\]

Answer 1

Given: \[ABCD\] is a quadrilateral. 
We need to find the sum of \[\overrightarrow{BA} + \overrightarrow{BC} + \overrightarrow{CD} + \overrightarrow{DA} .\]

Consider,
\[\overrightarrow{BA} + \overrightarrow{BC} + \overrightarrow{CD} + \overrightarrow{DA} \]
\[ = \left( \overrightarrow{BA} + \hspace{0.167em} \overrightarrow{DA} \right) + \left( \overrightarrow{BC} + \overrightarrow{CD} \right)\]
\[= \left( \overrightarrow{BD} + 2 \overrightarrow{DA} \right) + \overrightarrow{BD}\]                      [∵ \[\overrightarrow{BD} + \overrightarrow{DA} = \overrightarrow{BA}\] and \[\overrightarrow{BC} + \overrightarrow{CD} = \overrightarrow{BD}\]]

\[= 2 \left( \overrightarrow{BD} + \overrightarrow{DA} \right)\]

\[ = 2 \overrightarrow{BA}\]