Question

If $\overrightarrow{a}$ be the position vector whose tip is (5, −3), find the coordinates of a point B such that $\overrightarrow{AB}=$$\overrightarrow{a}$, the coordinates of A being (4, −1).

Answer 1

Let $O$ be the origin and let $P\left(5,-3\right)$ be the tip of the position vector $\overrightarrow{a}$. Then, $\overrightarrow{a}=\overrightarrow{OP}=5\hat{i}-3\hat{j}.$ Let the coordinate of $B$ be $\left(x,y\right)$ and $A$ has coordinates $\left(4,-1\right)$.
Therefore,
$\overrightarrow{AB}$ = Position vector of $B$ $-$ Position vector of $A$
$$\begin{gathered} =\left(x\hat{i}+y\hat{j}\right)-\left(4\hat{i}-\hat{j}\right) \\ =\left(x-4\right)\hat{i}+\left(y+1\right)\overline{j} \end{gathered}$$
Now,
$$\begin{gathered} \overrightarrow{AB}=\overrightarrow{a} \\ \Rightarrow \left(x-4\right)\hat{i}+\left(y+1\right)\hat{j}=5\hat{i}-3\hat{j} \\ \Rightarrow x-4=5\mathrm{and}y+1=-3 \\ \Rightarrow x=9\mathrm{and}y=-4 \end{gathered}$$
Hence, the coordinates of $B$ are $\left(9,-4\right)$.