Points A and B have position vectors a and b with respect to an origin O.
Points P,Q and R are defined such that 3OP = OA, 3OQ = 2OB, 2PR = RQ and S is the point of intersection of AB and OR produced.
Find the position vector of R.
If AS = kAB and OS = lOR find the values of k and l.
Hence write the position vector of S.
I have drawn the attached diagram and have worked out that the position vector of R is:
$\frac{1}{3}$( $\frac{2}{3}$a + $\frac{1}{3}$b) = $\frac{2}{9}$(a + b)
So S = $\frac{2l}{9}$(a + b)
But I can proceed no further.
Best Answer
Using $AS = kAB$, we have:
$$\overrightarrow{OS}= \mathbf a + k \overrightarrow{AB} = \mathbf a + k (\mathbf b - \mathbf a) = (1-k)\mathbf a +k \mathbf b$$
You already got $\overrightarrow{OS} = \dfrac{2l}9 (\mathbf a + \mathbf b)$
Equate both and evaluate $k$, $l$.