Find a displacement and prove a line intersection ratio

Q divides OP in ratio 2:1. Relative to A,

$\begin{aligned}&AQ\\ & =\frac{AO+2AP}{3}\\ & =\frac{-OA+2(OP-OA)}3\\ & =\frac{2OP-3OA}3\\ & =\frac{\mathbf b-2\mathbf a}{3}.\end{aligned}$

$OQ=\frac23OP=\frac13(\mathbf a+\mathbf b)$, so

$\begin{aligned}&AQ\\ & =OQ-OA\\ & =\frac13(\mathbf a+\mathbf b)-\mathbf a\\ & =\frac{\mathbf b-2\mathbf a}{3}.\end{aligned}$

Since O,R,B are collinear, write $OR=\lambda\mathbf b$; since A,Q,R are collinear, $AR=\mu AQ$. Then

$\lambda\mathbf b-\mathbf a=\frac\mu3(\mathbf b-2\mathbf a).$

Rearranging gives $(2\mu/3-1)\mathbf a=(\mu/3-\lambda)\mathbf b$. Independence of a,b requires $2\mu/3-1=0$ and $\mu/3-\lambda=0$. Thus $\mu=3/2$, $\lambda=1/2$, proving $OR=\mathbf b/2$.

Alternatively

$\begin{aligned}&OR\\ & =OA+AR\\ & =\mathbf a+\frac\mu3(\mathbf b-2\mathbf a)\\ & =\lambda\mathbf b.\end{aligned}$

Equating coefficients gives $\lambda-\mu/3=0$ and $1-2\mu/3=0$, yielding the same $\mu=3/2$, $\lambda=1/2$ and $OR=\mathbf b/2$.