3.4: Particle path and section formula in a triangle

A particle $P$ has position vector $\begin{pmatrix} 2 + 12t \\ 5 - 5t \end{pmatrix}$ at time $t$ seconds, $t \geqslant 0$.

Write down the initial position vector of $P$.

Find the speed of $P$.

Determine whether $P$ passes through the point with position vector $\begin{pmatrix} 158 \\ -48 \end{pmatrix}$.

The diagram shows triangle $OAC$. Point $B$ lies on $AC$ such that $AB:AC = 1:4$. Given $\overrightarrow{OA} = \mathbf{a}$, $\overrightarrow{OB} = \mathbf{b}$ and $\overrightarrow{OC} = \mathbf{c}$, find $\mathbf{c}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.