Question

If \[\vec{a} , \vec{b}\] are the vectors forming consecutive sides of a regular hexagon ABCDEF, then the vector representing side CD is 

Options

• \[\vec{a} + \vec{b}\]

   

• \[\vec{a} - \vec{b}\]

   

• \[\vec{b} - \vec{a}\]

   

• \[- \left( \vec{a} + \vec{b} \right)\]

   

Answer 1

\[\vec{b} - \vec{a}\]}
Let ABCDEF  be a regular hexagon such that \[\overrightarrow{AB} = \vec{a}\] and \[\overrightarrow{BC} = \vec{b} .\]
We know,
AD  is parallel to BC  such that AD = 2BC 
∴ \[\overrightarrow{AD} = 2 \overrightarrow{BC} = 2 \vec{b}\]
In \[\bigtriangleup ABC\], we have 
\[\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC} \]
\[ \Rightarrow \vec{a} + \vec{b} = \overrightarrow{AC}\] 
In \[\bigtriangleup ACD\], we have
\[\overrightarrow{AC} + \overrightarrow{CD} = \overrightarrow{AD} \]
\[ \Rightarrow \vec{CD} = \overrightarrow{AD} - \overrightarrow{AC} \]
\[ \Rightarrow \overrightarrow{CD} = 2 \vec{b} - \left( \vec{a} + \vec{b} \right)\]
\[ \Rightarrow \overrightarrow{CD} = \vec{b} - \vec{a}\]