2.5: Unit vector and points dividing line segments

Find the unit vector in the direction of $5\mathbf{i} - 15\mathbf{j}$.

The position vectors of points $A$ and $B$ relative to an origin $O$ are $\begin{pmatrix} 3 \\ -5 \end{pmatrix}$ and $\begin{pmatrix} 12 \\ 7 \end{pmatrix}$ respectively. The point $C$ lies on $AB$ such that $AC:CB$ is $2:1$.

Find the position vector of $C$ relative to $O$.

The point $D$ lies on $OB$ such that $OD:OB$ is $1:\lambda$ and $\overrightarrow{DC} = \begin{pmatrix} 6 \\ 1.25 \end{pmatrix}$.

Find the value of $\lambda$.