- \(A B C D\) is a parallelogram with \(A B\) parallel to \(D C\) and \(A D\) parallel to \(B C\). The position vectors of \(A , B , C\), and \(D\) relative to a fixed origin \(O\) are \(\mathbf { a } , \mathbf { b } , \mathbf { c }\) and \(\mathbf { d }\) respectively.
Given that
$$\mathbf { a } = \mathbf { i } + \mathbf { j } - 2 \mathbf { k } , \quad \mathbf { b } = 3 \mathbf { i } - \mathbf { j } + 6 \mathbf { k } , \quad \mathbf { c } = - \mathbf { i } + 3 \mathbf { j } + 6 \mathbf { k }$$
- find the position vector \(\mathbf { d }\),
- find the angle between the sides \(A B\) and \(B C\) of the parallelogram,
- find the area of the parallelogram \(A B C D\).
The point \(E\) lies on the line through the points \(C\) and \(D\), so that \(D\) is the midpoint of \(C E\).
- Use your answer to part (c) to find the area of the trapezium \(A B C E\).