Let vec{p} and vec{q} be the position vectors | vedclass

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(A) The position vectors of $R$ and $S$ are given by the section formula.For internal division in ratio $m:n = 2:3$,$\vec{OR} = \frac{n\vec{p} + m\vec

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$9p^{2} = 4q^{2}$

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(A) The position vectors of $R$ and $S$ are given by the section formula.For internal division in ratio $m:n = 2:3$,$\vec{OR} = \frac{n\vec{p} + m\vec{q}}{m+n} = \frac{3\vec{p} + 2\vec{q}}{5}$.For external division in ratio $m:n = 2:3$,$\vec{OS} = \frac{n\vec{p} - m\vec{q}}{n-m} = \frac{3\vec{p} - 2\vec{q}}{3-2} = 3\vec{p} - 2\vec{q}$.Given $\vec{OR} \perp \vec{OS}$,their dot product is zero: $\vec{OR} \cdot \vec{OS} = 0$.$\left(\frac{3\vec{p} + 2\vec{q}}{5}\right) \cdot (3\vec{p} - 2\vec{q}) = 0$.$9|\vec{p}|^2 - 6\vec{p}\cdot\vec{q} + 6\vec{q}\cdot\vec{p} - 4|\vec{q}|^2 = 0$.$9p^2 - 4q^2 = 0$.Therefore,$9p^2 = 4q^2$.

Let $\vec{p}$ and $\vec{q}$ be the position vectors of points $P$ and $Q$ with respect to the origin $O$,and $|\vec{p}| = p$,$|\vec{q}| = q$. If points $R$ and $S$ divide the line segment $PQ$ internally and externally in the ratio $2:3$ respectively,and if $\vec{OR} \perp \vec{OS}$,then:

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