7. The point \(A\) has position vector \(\mathbf { a } = 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k }\) and the point \(B\) has position vector \(\mathbf { b } = \mathbf { i } + \mathbf { j } - 4 \mathbf { k }\), relative to an origin \(O\).
- Find the position vector of the point \(C\), with position vector \(\mathbf { c }\), given by
$$\mathbf { c } = \mathbf { a } + \mathbf { b } .$$
- Show that \(O A C B\) is a rectangle, and find its exact area.
The diagonals of the rectangle, \(A B\) and \(O C\), meet at the point \(D\).
- Write down the position vector of the point \(D\).
- Find the size of the angle \(A D C\).