Question

In ∆OAB, E is the mid-point of OB and D is the point on AB such that AD : DB = 2 : 1. If OD and AE intersect at P, then determine the ratio OP : PD using vector methods.

Answer 1

Let A, B, D, E, P have position vectors \(\overline{a},\overline{b},\overline{d},\) \(\overline{e}, \overline{p}\) respectively w.r.t. O.

∵ AD : DB = 2 : 1.

∴ D divides AB internally in the ratio 2 : 1. 

Using section formula for internal division, we get

LHS is the position vector of the point which divides OD internally in the ratio 3 : 2. 

RHS is the position vector of the point which divides AE internally in the ratio 4 : 1. 

But OD and AE intersect at P 

∴ P divides OD internally in the ratio 3 : 2. 

Hence, OP : PD = 3 : 2.