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,
$$\begin{gathered} \overrightarrow{BA}+\overrightarrow{BC}+\overrightarrow{CD}+\overrightarrow{DA} \\ =\left(\overrightarrow{BA}+\hspace{0.17em}\overrightarrow{DA}\right)+\left(\overrightarrow{BC}+\overrightarrow{CD}\right) \end{gathered}$$
$=\left(\overrightarrow{BD}+2\overrightarrow{DA}\right)+\overrightarrow{BD}$ [∵ $\overrightarrow{BD}+\overrightarrow{DA}=\overrightarrow{BA}$ and $\overrightarrow{BC}+\overrightarrow{CD}=\overrightarrow{BD}$]
$$\begin{gathered} =2\left(\overrightarrow{BD}+\overrightarrow{DA}\right) \\ =2\overrightarrow{BA} \end{gathered}$$