Question

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two non-collinear vectors having the same initial point. What are the vectors represented by $\overrightarrow{a}$ + $\overrightarrow{b}$ and $\overrightarrow{a}$ − $\overrightarrow{b}$.

Answer 1

Given: $\overrightarrow{a},\overrightarrow{b}$ are two non-collinear vectors having same initial points. Complete the parallelogram $ABCD$ such that $\overrightarrow{AB}=\overrightarrow{a}$ and $\overrightarrow{BC}=\overrightarrow{b}.$
In $△ABC$,
$$\begin{gathered} \overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC} \\ \Rightarrow \overrightarrow{a}+\overrightarrow{b}=\overrightarrow{AC} \end{gathered}$$
In $△ABD$,
$$\begin{gathered} \overrightarrow{AD}+\overrightarrow{DB}=\overrightarrow{AB} \\ \Rightarrow \overrightarrow{b}+\overrightarrow{DB}=\overrightarrow{a} \\ \Rightarrow \overrightarrow{DB}=\overrightarrow{a}-\overrightarrow{b} \end{gathered}$$

Therefore, $\overrightarrow{AC}$ and $\overrightarrow{DB}$ are the diagonals of a parallelogram whose adjacent sides are $\overrightarrow{a}$ and $\overrightarrow{b}$ respectively.