10 The points \(A\) and \(B\) have position vectors \(2 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }\) and \(5 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\) respectively. The plane \(p\) has equation \(x + y = 5\).
- Find the position vector of the point of intersection of the line through \(A\) and \(B\) and the plane \(p\).
- A second plane \(q\) has an equation of the form \(x + b y + c z = d\), where \(b , c\) and \(d\) are constants. The plane \(q\) contains the line \(A B\), and the acute angle between the planes \(p\) and \(q\) is \(60 ^ { \circ }\). Find the equation of \(q\).