5 Points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\). Point \(C\) lies on \(A B\) such that \(A C : C B = p : 1\).
- Show that the position vector of \(C\) is \(\frac { 1 } { p + 1 } ( \mathbf { a } + p \mathbf { b } )\).
It is now given that \(\mathbf { a } = 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }\) and \(\mathbf { b } = - 6 \mathbf { i } + 4 \mathbf { j } + 12 \mathbf { k }\), and that \(C\) lies on the \(y\)-axis.
- Find the value of \(p\).
- Write down the position vector of \(C\).