\(OACB\) is a trapezium. \(\overrightarrow{OA}=2\mathbf a\), \(\overrightarrow{OB}=3\mathbf b\) and \(\overrightarrow{OA}=\dfrac23\overrightarrow{BC}\). \(X\) is the point on \(AB\) such that \(AX:XB=1:3\).
Express, as simply as possible, in terms of \(\mathbf a\) and/or \(\mathbf b\),
\(\overrightarrow{AX}\),[2]
\(\overrightarrow{OX}\).[1]
\(Y\) is the point on \(BC\) produced such that \(\overrightarrow{BY}=3\overrightarrow{OA}\). Find, in terms of \(\mathbf a\) and/or \(\mathbf b\), \(\overrightarrow{XY}\).[2]
Explain why \(O\), \(X\) and \(Y\) lie on a straight line.[2]
Find the ratio of the area of triangle \(OBX\) to the area of quadrilateral \(OACB\).[2]
\(W\) is the point on \(AB\) such that triangle \(OAW\) is similar to triangle \(CBW\). Find, in terms of \(\mathbf a\) and/or \(\mathbf b\), \(\overrightarrow{OW}\).[2]