Use your index finger to do those exercises:
1. Go from A to C: From A to B + from B to C. With vectors:
$\displaystyle \overrightarrow{AC} = \vec a + \vec b$
Since P is the midpoint of AC you'll get:
$\displaystyle \overrightarrow{AP} = \frac12 (\vec a + \vec b)$
2. According the method from #1 you'll get
$\displaystyle \overrightarrow{AQ} = \vec a + \frac12 \cdot \vec b$
3. You "go" from P to Q like: From P to A + from A to Q. With vectors:
$\displaystyle \overrightarrow{PQ} = -\frac12 \cdot ( \vec a + \vec b) + \vec a + \frac12 \cdot \vec b$
$\displaystyle \overrightarrow{PQ} = \frac12 \cdot \vec a $
4. This vector has the same direction as $\displaystyle \vec a$ and half of it's length.