Find the position vector of a point R which divides the line joining the two points P and Q with position vectors OP -2 a - b - and OQ - a -2 b - respectively in the ratio 1- 2 internally and externally- -NCERT EXEMPLAR-

It is given that P and Q are two points with position vectors OP #8594;=2a #8594;+b #8594; and OQ #8594;=a #8594; 2b #8594;, respectively. When R ...

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Standard XII

Mathematics

Relation between O,G,C

Find the posi...

Question

Find the position vector of a point R which divides the line joining the two points P and Q with position vectors $\vec{OP} = 2 \vec{a} + \vec{b}$ and $\vec{OQ} = \vec{a} - 2 \vec{b}$ , respectively in the ratio 1 : 2 internally and externally. [NCERT EXEMPLAR]

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Solution

It is given that P and Q are two points with position vectors $\vec{OP} = 2 \vec{a} + \vec{b}$ and $\vec{OQ} = \vec{a} - 2 \vec{b}$ , respectively.

When R divides PQ internally in the ratio 1 : 2, then

Position vector of R = $\frac{1 \times (\vec{a} - 2 \vec{b}) + 2 \times (2 \vec{a} + \vec{b})}{1 + 2} = \frac{5 \vec{a}}{3}$

When R divides PQ externally in the ratio 1 : 2, then

Position vector of R = $\frac{1 \times (\vec{a} - 2 \vec{b}) - 2 \times (2 \vec{a} + \vec{b})}{1 - 2} = \frac{- 3 \vec{a} - 4 \vec{b}}{- 1} = 3 \vec{a} + 4 \vec{b}$

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Find the position vector of a point R which divides the line joining the two points P and Q with position vectors OP→=2a→+b→ and OQ→=a→-2b→, respectively in the ratio 1 : 2 internally and externally. [NCERT EXEMPLAR]

Find the position vector of a point R which divides the line joining the two points P and Q with position vectors $\vec{OP} = 2 \vec{a} + \vec{b}$ and $\vec{OQ} = \vec{a} - 2 \vec{b}$ , respectively in the ratio 1 : 2 internally and externally. [NCERT EXEMPLAR]