Question

If $\overrightarrow{a},\overrightarrow{b}$are the vectors forming consecutive sides of a regular hexagon ABCDEF, then the vector representing side CD is
(a) $\overrightarrow{a}+\overrightarrow{b}$

(b) $\overrightarrow{a}-\overrightarrow{b}$

(c) $\overrightarrow{b}-\overrightarrow{a}$

(d) $-\left(\overrightarrow{a}+\overrightarrow{b}\right)$

Answer 1

(c) $\overrightarrow{b}-\overrightarrow{a}$
Let $ABCDEF$ be a regular hexagon such that $\overrightarrow{AB}=\overrightarrow{a}$ and $\overrightarrow{BC}=\overrightarrow{b}.$We know, $AD$ is parallel to $BC$ such that $AD=2BC$.
∴ $\overrightarrow{AD}=2\overrightarrow{BC}=2\overrightarrow{b}$
In $△ABC$, we have
$$\begin{gathered} \overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC} \\ \Rightarrow \overrightarrow{a}+\overrightarrow{b}=\overrightarrow{AC} \end{gathered}$$

In $△ACD$, we have
$$\begin{gathered} \overrightarrow{AC}+\overrightarrow{CD}=\overrightarrow{AD} \\ \Rightarrow \overrightarrow{CD}=\overrightarrow{AD}-\overrightarrow{AC} \\ \Rightarrow \overrightarrow{CD}=2\overrightarrow{b}-\left(\overrightarrow{a}+\overrightarrow{b}\right) \\ \Rightarrow \overrightarrow{CD}=\overrightarrow{b}-\overrightarrow{a} \end{gathered}$$