3 The position vector of point \(A\) is \(\mathbf { a } = - 9 \mathbf { i } + 2 \mathbf { j } + 6 \mathbf { k }\).
The line \(l\) passes through \(A\) and is perpendicular to \(\mathbf { a }\).
- Determine the shortest distance between the origin, \(O\), and \(l\).
\(l\) is also perpendicular to the vector \(\mathbf { b }\) where \(\mathbf { b } = - 2 \mathbf { i } + \mathbf { j } + \mathbf { k }\). - Find a vector which is perpendicular to both \(\mathbf { a }\) and \(\mathbf { b }\).
- Write down an equation of \(l\) in vector form.
\(P\) is a point on \(l\) such that \(P A = 2 O A\). - Find angle \(P O A\) giving your answer to 3 significant figures.
\(C\) is a point whose position vector, \(\mathbf { c }\), is given by \(\mathbf { c } = p \mathbf { a }\) for some constant \(p\). The line \(m\) passes through \(C\) and has equation \(\mathbf { r } = \mathbf { c } + \mu \mathbf { b }\). The point with position vector \(9 \mathbf { i } + 8 \mathbf { j } - 12 \mathbf { k }\) lies on \(m\). - Find the value of \(p\).