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 \[\vec{a}\] − \[\vec{b}\].

 

Answer 1

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,

\[\overrightarrow{AC}\] and \[\overrightarrow{DB}\] are the diagonals of a parallelogram whose adjacent sides are \[\vec{a}\] and \[\vec{b}\]  respectively.