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Misc 9 - Find R, if it divides P(2a + b), Q(a - 3b) externally - Section formula

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  1. Chapter 10 Class 12 Vector Algebra
  2. Serial order wise
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Misc 9 Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are (2 𝑎﷯ + 𝑏﷯) and ( 𝑎﷯ – 3 𝑏﷯) externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ. 𝑂𝑃﷯ = 2 𝑎﷯ + 𝑏﷯ 𝑂𝑄﷯ = 𝑎﷯ − 3 𝑏﷯ R divides PQ externally in the ratio 1 : 2 position vector of R = 𝟏 𝑶𝑸﷯ − 𝟐 𝑶𝑷﷯﷮𝟏 − 𝟐﷯ 𝑂𝑅﷯ = 1 𝑎﷯ − 3 𝑏﷯﷯ − 2(2 𝑎﷯ + 𝑏﷯)﷮−1﷯ = 𝑎﷯ − 3 𝑏﷯ − 4 𝑎﷯ − 2 𝑏﷯﷮−1﷯ = −3 𝑎﷯ − 5 𝑏﷯ ﷮−1﷯ = 𝟑 𝒂﷯+𝟓 𝒃﷯ So, position vector of R = 𝑂𝑅﷯ = 3 𝑎﷯+5 𝑏﷯ Now we have to find the mid point of RQ Position vector of mid-point = 𝑂𝑄﷯ + 𝑂𝑅﷯﷮2﷯ = 𝑎﷯ − 3 𝑏﷯ + 3 𝑎﷯ + 5 𝑏﷯﷮2﷯ = 4 𝑎﷯ + 2 𝑏﷯﷮2﷯ = 2 𝑎﷯+ 𝑏﷯ This is the position vector of P. So, P is the mid point of RQ.

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