Let \({\rm{\vec p}}\) and \({\rm{\vec q}}\) be the position vectors of the points P and Q respectively with respect to origin O. The points r and S divide PQ internally and externally respectively in the ratio 2 : 3 If \(\overrightarrow {{\rm{OR}}} \) and \(\overrightarrow {{\rm{OS}}} \) are perpendicular, then which one of the following is correct?

Concept:

1. Let the given two position vector p and q respectively and r be the vector dividing the segment PQ internally in the ratio m: n

1. Internal Section Formula: When the segment PQ is divided internally in the ration m: n, we use this formula. ⇔ \({\rm{r}} = \left( {\frac{{{\rm{mq\;}} + {\rm{\;np}}}}{{{\rm{m}} + {\rm{\;n}}}}} \right)\)
2. External Section Formula: When the segment PQ is divided externally in the ratio m: n, we use this formula. ⇔ \({\rm{s}} = \left( {\frac{{{\rm{mq}} - {\rm{np}}}}{{{\rm{m}} - {\rm{\;n}}}}} \right)\)

2. \({\rm{\vec a\;and\;\vec b}}\) are two vectors perpendicular to each other ⇔ \({\rm{\vec a}}.{\rm{\vec b}} = 0\)

Calculation:

Given that,

Vector R divide segment PQ internally in ration 2 : 3 so,

Form the diagram,

\(\Rightarrow {\rm{R}} = \left( {\frac{{2{\rm{q\;}} + {\rm{\;}}3{\rm{p}}}}{{2 + {\rm{\;}}3}}} \right)\)

\(\Rightarrow {\rm{R}} = \frac{{\left( {2{\rm{q}} + 3{\rm{p}}} \right)}}{5}\)       …(1)

Given Vector S divide segment PQ internally in ration 2 : 3 so,

\(\Rightarrow {\rm{S}} = \left( {\frac{{2{\rm{q}} - 3{\rm{p}}}}{{2 - {\rm{\;}}3}}} \right)\)

\(\Rightarrow {\rm{S}} = \frac{{3{\rm{p}} - 2{\rm{q}}}}{1}\)       …(2)

Note: given vectors are position vectors with respect to O so vector OR = R and OS = s

Given OR and OS perpendicular to each other so,

⇒ R.S = 0

From equation 1 and 2

\(\Rightarrow \frac{{\left( {2{\rm{q}} + 3{\rm{p}}} \right)}}{5}.\frac{{3{\rm{p}} - 2{\rm{q}}}}{1} = 0\)

⇒ 6q.p - 4q² + 9p² - 6p.q = 0

⇒ 4q² = 9p²