N1987 P1 Q18 [Modified]

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]