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for internal and external division

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Let P and Q be two points represented by the position vectors `vec (OP)` and `vec (OQ)`, respectively, with respect to the origin O. Then the line segment joining the points P and Q may be divided by a third point, say R, in two ways – internally and externally. Then we find the position vector `vec (OR)` for the point R with respect to the origin O . We take the two cases one by one.

Case I - Fig.

 When R divides PQ internally . If R divides `vec (PQ)` such that m `vec (RQ)` = n `vec (PR)`, where m and n are positive scalars, we say that the point R divides `vec (PQ)` internally in the ratio of m : n. Now  from triangles ORQ and OPR, we have

`vec (RQ) = vec (OQ) - vec(OR) = vec b - vec r` and 

`vec (PR) = vec (OR) - vec (OP) = vec r - vec a`, Therefore we have 

`m(vec b - vec r) = n (vec r - vec a),`

`vec r = (m vec b + n vec a)/(m+n)`           ( On simplification)
Hence, the position vector of the point R which divides P and Q internally in the ratio of m : n is given by
`vec (OR) = (m vec b + n vec a)/(m + n)`

Case II - When R divides PQ externally Fig.


The position vector of the point R which divides the line segment PQ externally in the ratio m : n i.e. `(PR)/(QR) = m/n` is given by 
`vec (OR) = (m vec b - n vec a)/(m - n)`

Remark:
If R is the midpoint of PQ , then m = n. And therefore, from Case I, the midpoint R of `vec (PQ)`, will have its position vector as 
`vec (OR) = (vec a + vec b)/2`   

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