5. Relative to a fixed origin \(O\), the point \(A\) has position vector \(3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\), the point \(B\) has position vector \(5 \mathbf { i } + \mathbf { j } + \mathbf { k }\), and the point \(C\) has position vector \(7 \mathbf { i } - \mathbf { j }\).

1. Find the cosine of angle \(A B C\).
2. Find the exact value of the area of triangle \(A B C\). The point \(D\) has position vector \(7 \mathbf { i } + 3 \mathbf { k }\).
3. Show that \(A C\) is perpendicular to \(C D\).
4. Find the ratio \(A D : D B\).