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Question
If \[\vec{a}\] and \[\vec{b}\] are two non-collinear vectors having the same initial point. What are the vectors represented by \[\vec{a}\] + \[\vec{b}\] and \[\vec{a}\] − \[\vec{b}\].
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Solution
Given: \[\vec{a} , \vec{b}\] are two non-collinear vectors having same initial points. Complete the parallelogram \[ABCD\] such that \[\overrightarrow{AB} = \vec{a}\] and \[\overrightarrow{BC} = \vec{b} .\]
In \[\bigtriangleup ABC\]
\[\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC} \]
\[ \Rightarrow \vec{a} + \vec{b} = \overrightarrow{AC}\]
In \[\bigtriangleup ABD\]
\[\overrightarrow{AD} + \overrightarrow{DB} = \overrightarrow{AB} \]
\[ \Rightarrow \vec{b} + \overrightarrow{DB} = \vec{a} \]
\[ \Rightarrow \overrightarrow{DB} = \vec{a} - \vec{b}\]
Therefore,
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