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(A) The position vector of a point $R$ that divides the line segment joining points $P$ and $Q$ with position vectors $\vec{p}$ and $\vec{q}$ external

Vedclass · Accepted Answer

$4\vec{b} - \vec{a}$

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(A) The position vector of a point $R$ that divides the line segment joining points $P$ and $Q$ with position vectors $\vec{p}$ and $\vec{q}$ externally in the ratio $m:n$ is given by the formula $\overrightarrow{OR} = \frac{m\vec{q} - n\vec{p}}{m - n}$.Given $\vec{p} = 3\vec{a} - 2\vec{b}$,$\vec{q} = \vec{a} + \vec{b}$,$m = 2$,and $n = 1$.Substituting these values into the formula:$\overrightarrow{OR} = \frac{2(\vec{a} + \vec{b}) - 1(3\vec{a} - 2\vec{b})}{2 - 1}$$\overrightarrow{OR} = \frac{2\vec{a} + 2\vec{b} - 3\vec{a} + 2\vec{b}}{1}$$\overrightarrow{OR} = (2\vec{a} - 3\vec{a}) + (2\vec{b} + 2\vec{b})$$\overrightarrow{OR} = 4\vec{b} - \vec{a}$.

Consider two points $P$ and $Q$ with position vectors $\overrightarrow{OP} = 3\vec{a} - 2\vec{b}$ and $\overrightarrow{OQ} = \vec{a} + \vec{b}$. Find the position vector of a point $R$ which divides the line joining $P$ and $Q$ in the ratio $2:1$ externally.

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