1.1: Vectors and an area ratio

Figure 4 shows triangle $\displaystyle OAB$ in which

The point $\displaystyle P$ lies on $\displaystyle AB$ such that $\displaystyle AP:PB=3:1$.

The point $\displaystyle M$ is the midpoint of $\displaystyle OA$ and the point $\displaystyle N$ is the midpoint of $\displaystyle OP$.

(a) Find, as simplified expressions in terms of $\displaystyle \mathbf{a}$ and $\displaystyle \mathbf{b}$, the vector

(i) $\displaystyle \overrightarrow{OP}$,

(ii) $\displaystyle \overrightarrow{MN}$.

The point $\displaystyle C$ lies on $\displaystyle OB$ such that $\displaystyle ANC$ is a straight line.

(b) Using a vector method, find $\displaystyle \overrightarrow{OC}$ as a simplified expression in terms of $\displaystyle \mathbf{b}$.

Given that

(c) find the exact value of $\displaystyle K$.