Vector Properties in Geometric Shapes: The Parallelogram

In the diagram, \(OABC\) is a parallelogram. \(D\) is the mid-point of \(BC\). It is given that \(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OC} = \mathbf{c}\). \(E\) is the point on \(OC\) such that \(5OE = 2OC\). \(F\) is the point on \(OD\) such that \(\overrightarrow{OF} = \frac{1}{2}\overrightarrow{FD}\).

1. Express, as simply as possible, in terms of \(\mathbf{a}\) and/or \(\mathbf{c}\),
   1. \(\overrightarrow{EC}\),
   2. \(\overrightarrow{OD}\),
   3. \(\overrightarrow{OF}\).
2. Show that \(\overrightarrow{AE} = 6\overrightarrow{FE}\).
3. Given that the coordinates of \(F\) are \((4,2)\) and \(\overrightarrow{AE} = \begin{pmatrix} 3 \\ -12 \end{pmatrix}\).
   1. Calculate \(|\overrightarrow{AE}|\).
   2. Find the coordinates of \(E\).
   3. Find the value of \(t\), given that the vector \(\begin{pmatrix} 4.5 \\ t \end{pmatrix}\) is parallel to \(\overrightarrow{AE}\).