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Question

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are externally in the ratio 1: 2. Also, show that P is the mid point of the line segment RQ.

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Solution

It is given that the point R divides the line joining two points, P with position vector 2 a + b and Q with position vector a 3 b externally in the ratio 1:2, that is O P =2 a + b and O Q = a 3 b .

As Rdivides PQexternally in the ratio 1:2, therefore by section formula,

OR = 2( 2 a + b )1( a 3 b ) 21 OR = 4 a +2 b a +3 b 1 OR =3 a +5 b

The midpoint of RQ is,

midpointof RQ = OQ + OR 2 = a 3 b +3 a +5 b 2 = 4 a +2 b 2 =2 a + b

The midpoint of RQ =2 a + b which is equal to the position vector of P.

Hence, Pis the midpoint of RQ .


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