1.1: Midpoint vector and vectors from magnitude and direction

The diagram shows a figure $OABC$, where $\overrightarrow{OA} = \mathbf{a}$, $\overrightarrow{OB} = \mathbf{b}$ and $\overrightarrow{OC} = \mathbf{c}$. The lines $AC$ and $OB$ intersect at the point $M$ where $M$ is the midpoint of the line $AC$.

Find, in terms of $\mathbf{a}$ and $\mathbf{c}$, the vector $\overrightarrow{OM}$.

Given that $OM:MB = 2:3$, find $\mathbf{b}$ in terms of $\mathbf{a}$ and $\mathbf{c}$.

Vectors $\mathbf{i}$ and $\mathbf{j}$ are unit vectors parallel to the $x$-axis and $y$-axis respectively. The vector $\mathbf{p}$ has a magnitude of $39$ units and has the same direction as $-10\mathbf{i} + 24\mathbf{j}$.

Find $\mathbf{p}$ in terms of $\mathbf{i}$ and $\mathbf{j}$.

Find the vector $\mathbf{q}$ such that $2\mathbf{p} + \mathbf{q}$ is parallel to the positive $y$-axis and has a magnitude of $12$ units.

Hence show that $|\mathbf{q}| = k\sqrt{5}$, where $k$ is an integer to be found.