The points O; A and B do not lie in a straight line.

OA = a and OB = b.

P is the mid-point of OA. Q lies on the line segment OB, such that

OQ : OB = 1 : 3. AQ and BP intersect at R.

(a) Express the vectors

AB; AQ; PB and PQ in terms of a and b.

explain why there are scalars $\displaystyle \lambda $ and $\displaystyle \mu $ such that

$\displaystyle RQ = \lambda AQ $

and $\displaystyle PR = \mu PB $

Hence find an expression for PQ in terms of a; b; $\displaystyle \lambda , \mu $ and

By equating the coecients of a and of b in two expressions for vector PQ, find $\displaystyle \lambda , \mu $ and Hence show that AR : RQ = 3 : 2 and find the ratio P R : RB.

Can someone check to see if these are correct?

AB = -a + b

AQ = -a + 1/3b

PB = 1/2a + b

PQ = 1/2a + 1/3b

I am bit stuck on the second part, I know their are scalars, because the both vectors are the same, but have different magnitudes?