\displaystyle \textbf{Question 1:} \text{Find the position vector of a point }R\text{ which divides the line segment joining } \\ P\text{ and }Q\text{ whose position vectors are }2\overrightarrow{a}+\overrightarrow{b}\text{ and }\overrightarrow{a}-3\overrightarrow{b},\text{ externally in the ratio }\\ 1:2. \text{ Also, show that }P\text{ is the mid-point of the line segment }RQ. \text{[CBSE 2010]}

\displaystyle \text{Answer:}
\displaystyle \text{It is given that }R\text{ divides }PQ\text{ externally in the ratio }1:2.
\displaystyle \therefore\ \text{The position vector of }R =\frac{1\times(\overrightarrow{a}-3\overrightarrow{b})-2(2\overrightarrow{a}+\overrightarrow{b})}{1-2} =3\overrightarrow{a}+5\overrightarrow{b}
\displaystyle \text{Now,}
\displaystyle \frac{\text{Position vector of }R+\text{Position vector of }Q}{2} =\frac{3\overrightarrow{a}+5\overrightarrow{b}+\overrightarrow{a}-3\overrightarrow{b}}{2}
\displaystyle =2\overrightarrow{a}+\overrightarrow{b} =\text{Position vector of }P
\displaystyle \text{Hence, }P\text{ is the mid-point of }PQ.

\displaystyle \textbf{Question 2:} \text{The two vectors }\widehat{i}+\widehat{i}\text{ and }3\widehat{i}-\widehat{j}+4\widehat{k}\text{ represent the sides }AB\text{ and }AC \\ \text{ respectively of triangle }ABC.\ \text{Find the length of the median through }A. \text{[CBSE 2015, 2016]}

\displaystyle \text{Answer:}
\displaystyle \text{Let }D\text{ be the mid-point of side }BC\text{ of triangle }ABC.\ \text{Then,}
\displaystyle \overrightarrow{AB}+\overrightarrow{AC}=2\overrightarrow{AD}
\displaystyle \Rightarrow (\widehat{j}+\widehat{i})+(3\widehat{i}-\widehat{j}+4\widehat{k}) =2\overrightarrow{AD}
\displaystyle \Rightarrow \overrightarrow{AD}=2\widehat{i}+0\widehat{j}+2\widehat{k}
\displaystyle \Rightarrow \left|\overrightarrow{AD}\right| =\sqrt{4+0+4}=2\sqrt{2}

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