Recover a dot product, projection and a parameterised area

Referred to the origin O, the points A and B have position vectors a and b respectively such that $|\mathbf a|=5$ and $|\mathbf b|=3$.

(i) It is given that $|2\mathbf b-\mathbf a|=7$. Using a suitable scalar product, show that $\mathbf a\cdot\mathbf b=3$.2 marks

(ii) Deduce the length of projection of $\overrightarrow{OA}$ onto $\overrightarrow{OB}$.1 mark

The point P lies on AB such that AP:PB=1:k, where k is a positive constant.

(iii) Write down the position vector of P in terms of k,a and b and show that the area of triangle OAP is $\dfrac{3\sqrt6}{k+1}$.5 marks