Connect a cross-product relation to areas and a nonzero linear combination

Relative to the origin $O$, three distinct fixed points $A$, $B$ and $C$ have position vectors $\mathbf a$, $\mathbf b$ and $\mathbf c$ respectively. It is known that $\mathbf b$ is a unit vector, $|\mathbf a|=3$, $|\mathbf c|=2$ and the angle $AOC$ is $60^\circ$.

(i) State the geometrical interpretation of $|\mathbf b\cdot\mathbf c|$.1 mark

It is further given that $\mathbf a\times2\mathbf b=\mathbf b\times\mathbf c$.

(ii) Find the ratio of the area of triangle $AOB$ to the area of triangle $BOC$.2 marks

(iii) Show that $2\mathbf a+\mathbf c=k\mathbf b$ where $k\in\mathbb R$, $k\ne0$.

By considering $(2\mathbf a+\mathbf c)\cdot(2\mathbf a+\mathbf c)$, find the exact values of $k$.6 marks