1987 TYS

The points $A$, $B$, $C$, $D$ have position vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, $\mathbf{d}$ given by $\mathbf{a}=\mathbf{i}+2\mathbf{j}+3\mathbf{k}$, $\mathbf{b}=\mathbf{i}+2\mathbf{j}+2\mathbf{k}$, $\mathbf{c}=3\mathbf{i}+2\mathbf{j}+\mathbf{k}$, $\mathbf{d}=4\mathbf{i}-\mathbf{j}-\mathbf{k}$, respectively. The point $P$ lies on $AB$ produced such that $AP=2AB$, and the point $Q$ is the midpoint of $AC$.

1. Show that $PQ$ is perpendicular to $AQ$.

   [4]

2. Find the area of the triangle $APQ$.

   [2]

3. Find a vector perpendicular to the plane $ABC$.

   [3]

4. Find the cosine of the angle between $\overrightarrow{AD}$ and $\overrightarrow{BD}$.

   [3]