Question

X and Y are two points with position vectors $3\overrightarrow{a}+\overrightarrow{b}\mathrm{and}\overrightarrow{a}-3\overrightarrow{b}$ respectively. Write down the position vector of a point Z which divides the line segment XY in the ration 2:1 externally.

Answer 1

Given:
X and Y are two points with position vectors $3\overrightarrow{a}+\overrightarrow{b}\mathrm{and}\overrightarrow{a}-3\overrightarrow{b}$, respectively.
Z divides the line segment XY in the ratio 2 : 1 externally

We know,
The position vector of R which divided P and Q externally in the ratio m : n is given by $\overrightarrow{OR}=\frac{m\overrightarrow{b}-n\overrightarrow{a}}{m-n}$, where position vector of P and Q are $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$, respectively.

Thus,
$$\begin{gathered} \overrightarrow{OZ}=\frac{2\left(\overrightarrow{a}-3\overrightarrow{b}\right)-1\left(3\overrightarrow{a}+\overrightarrow{b}\right)}{2-1} \\ \Rightarrow \overrightarrow{OZ}=\frac{2\overrightarrow{a}-6\overrightarrow{b}-3\overrightarrow{a}-\overrightarrow{b}}{1} \\ \Rightarrow \overrightarrow{OZ}=-\overrightarrow{a}-7\overrightarrow{b} \end{gathered}$$

Hence, the position vector of a point Z is $-\overrightarrow{a}-7\overrightarrow{b}.$