7 The points \(A\) and \(B\) have coordinates \(( 1,4,2 )\) and \(( 2 , - 1,3 )\) respectively.
The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 2
- 1
3 \end{array} \right] + \lambda \left[ \begin{array} { r } 1
- 1
1 \end{array} \right]\).

1. Show that the distance between the points \(A\) and \(B\) is \(3 \sqrt { 3 }\).
2. The line \(A B\) makes an acute angle \(\theta\) with \(l\). Show that \(\cos \theta = \frac { 7 } { 9 }\).
3. The point \(P\) on the line \(l\) is where \(\lambda = p\).
   1. Show that $$\overrightarrow { A P } \cdot \left[ \begin{array} { r } 1
      - 1
      1 \end{array} \right] = 7 + 3 p$$
   2. Hence find the coordinates of the foot of the perpendicular from the point \(A\) to the line \(l\).