N2023 P2 Q3

The points \(A\) and \(B\) have position vectors \(\left( \begin{matrix} -1 \\ 2 \\ 5 \\ \end{matrix} \right)\) and \(\left( \begin{matrix} 1 \\ -2 \\ 8 \\ \end{matrix} \right)\) respectively. The point \(C\) is such that \(\overrightarrow{BC}=2\overrightarrow{AB}\).

1. Find the position vector of \(C\).[2]

The point \(D\) has position vector \(\left( \begin{matrix} 1 \\ 2 \\ d \\ \end{matrix} \right)\) and is such that \(\left| \overrightarrow{AD} \right|=\left| \overrightarrow{BD} \right|\).

2. Find the value of \(d\).[2]
3. Use a scalar product to find angle \(ADB\).[3]
4. Find exactly the position vector of the point \(P\), where \(P\) is the centre of the circle that passes through \(A\),\(B\) and \(D\).[5]