Vector Applications: Ratios of Areas in Geometric Figures

In the diagram, \(AQ = 3QC\), \(QB = 5QX\), and \(P\) is the mid-point of \(AB\). \(\overrightarrow{AQ} = 3\mathbf{a}\) and \(\overrightarrow{AP} = \mathbf{b}\).

Triangle ABC with points P, Q, and intersection X

1. Express, as simply as possible, in terms of \(\mathbf{a}\) and or \(\mathbf{b}\),
   1. \(\overrightarrow{BQ}\),
   2. \(\overrightarrow{QX}\),
   3. \(\overrightarrow{CX}\),
   4. \(\overrightarrow{CP}\).
2. Calculate the value of
   1. \(\frac{CX}{CP}\),
   2. \(\frac{\text{area of}\hspace{0.5em} \Delta BXC}{\text{area of}\hspace{0.5em} \Delta BPX}\).