If $$\overrightarrow a $$ is a position vector of a point $$\left( {1,\, - 3} \right)$$  and $$A$$ is another point $$\left( { - 1,\,5} \right),$$   then what are the coordinates of the point $$B$$ such that $$\overrightarrow {AB} = \overrightarrow a \,?$$

Solution :
$$\eqalign{ & {\text{Let the coordinates of }}B{\text{ be }}\left( {x,{\text{ }}y} \right) \cr & \overrightarrow a = i - 3j \cr & {\text{P}}{\text{.V}}{\text{. of }}A{\text{ is}}\left( { - 1,\,5} \right) \cr & {\text{So,}}\,\overrightarrow {OA} = i + 5j,\,\overrightarrow {OB} = xi + yj \cr & \therefore \,\overrightarrow {AB} = \overrightarrow {OB} - \overrightarrow {OA} = \overrightarrow a \cr} $$
$$\Rightarrow \left(x+1\right)\stackrel{⌢}{i}+\left(y-5\right)\stackrel{⌢}{j}=\stackrel{⌢}{i}-3\stackrel{⌢}{j}$$
$$\eqalign{ & \Rightarrow x + 1 = 1{\text{ and }}y - 5 = - 3 \cr & \Rightarrow x = 0{\text{ and }}y = 2 \cr & \therefore \,{\text{Coordinates of }}B{\text{ are }}\left( {0,\,2} \right) \cr} $$