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Question
If \[\vec{a}\] and \[\vec{b}\] represent two adjacent sides of a parallelogram, then write vectors representing its diagonals.
Solution
Let \[\vec{a}\] and \[\vec{b}\] represents two adjacent sides of a parallelogram ABCD.
∴ \[AB = DC \text{ and }AD \hspace{0.167em} = BC\]
\[\Rightarrow \overrightarrow{DC} = \overrightarrow{AB} = \vec{a}\] and
\[\overrightarrow{AD} = \overrightarrow{BC} = \vec{b}\]
In \[\bigtriangleup ABC\]
\[\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC} \]
\[ \Rightarrow \vec{a} + \vec{b} = \overrightarrow{AC}\]
In \[\bigtriangleup ABD\]
\[\overrightarrow{AD} + \overrightarrow{DB} = \overrightarrow{AB} \]
\[ \Rightarrow \vec{b} + \overrightarrow{DB} = \vec{a} \]
\[ \Rightarrow \overrightarrow{DB} = \vec{a} - \vec{b}\]
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