Find a displacement and prove a line intersection ratio

Relatively to an origin O, the position vectors of A, B and P are $\mathbf a$, $\mathbf b$ and $\frac12(\mathbf a+\mathbf b)$ respectively, where a and b are non-zero and non-parallel vectors. The point Q lies on the line segment OP such that OQ:QP=2:1. The line AQ cuts OB at R.

(i) Find, in terms of a and b, $\overrightarrow{AQ}$.2 marks

(ii) Show that $\overrightarrow{OR}=\frac12\mathbf b$.3 marks