====== Question 11, Exercise 3.2 ======
Solutions of Question 11 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

=====Question 11(i)=====
Find the position vectors of the point of division of the line segments joining point $C$ with position vector $5\hat{j}$ and point $D$ with position vector $4\hat{i}+\hat{j}$ in the ratio $2:5$ internally. 
====Solution====
Position vector of $C$ is $\overrightarrow{OC}=5\hat{j}$

Position vector of $D$ is $\overrightarrow{OD}=4\hat{i}+\hat{j}$ 

Let $H$ be the point divides the line segment $\overline{CD}$ in the ratio $2:5$internally,

then by ratio theorem, we have position vector $H$ is:
\begin{align}\overrightarrow{OH}&=\dfrac{5\overrightarrow{OC}+2\overrightarrow{OD}}{5+2}\\  
&=\dfrac{5(5\hat{j})+2(4\hat{i}+\hat{j})}{7}\\
&=\dfrac{1}{7}(8\hat{i}+27\hat{j})\\
\implies \overrightarrow{OH}&=\dfrac{8}{7}\hat{i}+\dfrac{27}{7}\hat{j}\end{align}

=====Question 11(ii)=====
Find the position vectors of the point of division of the line segments joining point $E$ with position vector $2\hat{i}-3\hat{j}$ and point $F$ with position vector $3\hat{i}+2\hat{j}$ in the ratio $4:3$ externally. 
====Solution====
Position vector of $E$ is $\overrightarrow{OE}=2\hat{i}-3\hat{j}$

Position vector of $F$ is $\overrightarrow{OF}=3\hat{i}+2\hat{j}$ 

Let $K$ be the point with position vector $\overrightarrow{OK}$ that divides the line segment $\overline{EF}$ externally in the ratio $4:3$,
then by ratio theorem,
\begin{align}\overrightarrow{OK}&=\dfrac{3\overrightarrow{OE}-4\overrightarrow{OF}}{3-4}\\  
&=-[3(2\hat{i}-3\hat{j})-4(3\hat{i}+2\hat{j})]\\
&=-(6-12)\hat{i}-(-9-8)\hat{j}\\
\implies \overrightarrow{OK}&=6\hat{i}+17\hat{j}\end{align}


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