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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 $\vec{q}$ in the r

Vedclass · Accepted Answer

$\frac{5\vec{a}}{3}$

Vedclass · Answer

(A) The position vector of a point $R$ that divides the line segment joining points $P$ and $Q$ with position vectors $\vec{p}$ and $\vec{q}$ in the ratio $m:n$ internally is given by the section 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}$$\overrightarrow{OR} = \frac{2\vec{a} + 2\vec{b} + 3\vec{a} - 2\vec{b}}{3}$$\overrightarrow{OR} = \frac{5\vec{a}}{3}$.

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 segment joining $P$ and $Q$ in the ratio $2:1$ internally.

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