Let ABC be a triangle and be a , b , c the position vectors of A , B , C respectively. Let D divide BC in the ratio 3 : 1 internally and E divide AD in the ratio 4 : 1 internally. Let BE meet AC in F

Question

Let ABC be a triangle and be a , b , c the position vectors of A , B , C respectively. Let D divide BC in the ratio 3 : 1 internally and E divide AD in the ratio 4 : 1 internally. Let BE meet AC in F . IfE divides BF in the ratio 3 : 2 internally then the position vector of F is

TestHub · Accepted Answer

Here we are given that OA = a , OB = b , OC = c Now P.V of Di.e OD = 1 + 3 1 × OB + 3 × OC ​ ​ OD = 4 b + 3 c ​ OE = 4 + 1 4 OD + OA ​ = 5 4 4 ( b + 3 c ) ​ + a ​ ⇒ OE = 5 a + b + 3 c ​ Now, OE = 2 + 3 2 OB + 3 OF ​ ⇒ OF = 3 5 OE − 2 OB ​ ⇒ OF = 3 5 5 ( a + b + 3 c ) ​ − 2 b ​ ⇒ OF = 3 a − b + 3 c ​ ⇒ P . V . of F is 3 a − b + 3 c ​ ​

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