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.

Answer 1

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 → ) 2−1 OR → = 4 a → +2 b → − a → +3 b → 1 OR → =3 a → +5 b →

The midpoint of RQ → is,

midpoint of  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 → .