7. Relative to a fixed origin \(O\), the position vectors of the points \(A , B\) and \(C\) are $$\overrightarrow { O A } = - 3 \mathbf { i } + \mathbf { j } - 9 \mathbf { k } , \quad \overrightarrow { O B } = \mathbf { i } - \mathbf { k } , \quad \overrightarrow { O C } = 5 \mathbf { i } + 2 \mathbf { j } - 5 \mathbf { k } \text { respectively. }$$

1. Find the cosine of angle \(A B C\). The line \(L\) is the angle bisector of angle \(A B C\).
2. Show that an equation of \(L\) is \(\mathbf { r } = \mathbf { i } - \mathbf { k } + t ( \mathbf { i } + 2 \mathbf { j } - 7 \mathbf { k } )\).
3. Show that \(| \overrightarrow { A B } | = | \overrightarrow { A C } |\). The circle \(S\) lies inside triangle \(A B C\) and each side of the triangle is a tangent to \(S\).
4. Find the position vector of the centre of \(S\).
5. Find the radius of \(S\).