2020 HCI P2 Q2

\(6\) equilateral triangles each with sides \(5\) units are arranged to form a regular hexagon as shown in the diagram below. By expressing \(\overrightarrow{OQ}\) in terms of \(\overrightarrow{OP}\) and \(\overrightarrow{OR}\), or otherwise, find \(\overrightarrow{OP\cdot }\,\left( \overrightarrow{OP}+\overrightarrow{OQ}+\overrightarrow{OR} \right)\).[3]

\(\overrightarrow{OE}\),\(\overrightarrow{OF}\) and \(\overrightarrow{OG}\) are non-zero position vectors relative to a fixed origin \(O\) for points \(E\),\(F\) and \(G\) respectively. It is given that \(2\overrightarrow{OE}-5\overrightarrow{OF}+3\overrightarrow{OG}=0\). Show that \(E\), \(F\) and \(G\) are collinear. State the ratio of \(EF:EG\) and find \(\overrightarrow{EF}\times \overrightarrow{EG}\).[4]