3.27: Scaled vector, section formula and components

Find the vector with magnitude $200$ in the direction of $\begin{pmatrix} 7 \\ -24 \end{pmatrix}$.

The diagram shows triangle $AOB$ with $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. Point $C$ lies on $AB$ such that $AC:AB = 1:3$. Find $\overrightarrow{OC}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

Given $p\begin{pmatrix} 2 \\ 1 \end{pmatrix} + q\begin{pmatrix} 2 \\ 4 \end{pmatrix} = \begin{pmatrix} -p + 1 \\ 5 \end{pmatrix}$, find $p$ and $q$.