3 The points \(A\) and \(B\) have position vectors \(\mathbf { a } = \mathbf { i } + \mathrm { pj } + \mathrm { q } \mathbf { k }\) and \(\mathbf { b } = 2 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\) respectively, relative to the origin \(O\).
- Determine the value of \(p\) and the value of \(q\) for which \(\mathbf { a } \times \mathbf { b } = 2 \mathbf { i } + 6 \mathbf { j } - 1 \mathbf { 1 } \mathbf { k }\).
- The point \(C\) has coordinates ( \(d , e , f\) ) and the tetrahedron \(O A B C\) has volume 7.
- Using the values of \(p\) and \(q\) found in part (a), find the possible relationships between \(d , e\) and \(f\).
- Explain the geometrical significance of these relationships.