2023 DHS Promo Q5

With reference to the origin \(O\), the points \(A\) and \(B\) have coordinates \(\left( 1,2,0 \right)\) and \(\left( -5,1,0 \right)\).

1. Find the angle \(AOB\).[3]

The vectors \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) are projected along the direction of \(\left( \begin{matrix} 1 \\ 2 \\ 2 \\ \end{matrix} \right)\) and \(\left( \begin{matrix} 2 \\ 1 \\ 2 \\ \end{matrix} \right)\) to form the vectors \(\overrightarrow{O'A'}\) and \(\overrightarrow{O'B'}\).[3]

2. Find the vectors \(\overrightarrow{O'A'}\) and \(\overrightarrow{O'B'}\).[3]
3. Find the exact area of the triangle \(O'A'B'\).[2]