In a `triangle 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 of `OP: PD` using vector methods

Explanation

With O as origin let `veca and vecb` be the position vectors of A and B, respectively.

Then the position vector of E, the midpoint of OB, is `vecb//2`.
Again since `AD : DB = 2:1`, the position vecor of D is
`" "(1*veca + 2vecb)/(1+2) = (veca + 2vecb)/(3)`
Let `" "(OP)/(OD) = (1)/(lamda)`
`rArr ` P.V. of P `= (veca + 2 vecb)/(3(lamda +1))`
Let `(AP)/(PE) = (1)/(mu)`
`rArr ` P.V. of `P = (muveca + (vecb)/(2))/(mu+1)`
Comparing P.V. of P, we have
`" "(1)/(3(lamda +1)) = (mu)/(mu+1) and (2)/(3(lamda +1)) = (1)/(2(mu+1))`
Dividing `mu = (1)/(4) rArr lamda = (2)/(3)`
`rArr " "(OP)/(PA) = (3)/(2)`