2024 NYJC Promo Q5

The position vectors of points \(A\), \(B\) and \(C\), relative to the origin \(O\) are \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively, where

\[\mathbf{a} = 3\mathbf{i} + 2\mathbf{j} - \mathbf{k},\quad \mathbf{b} = 4\mathbf{i} - 3\mathbf{j} + 2\mathbf{k},\quad \mathbf{c} = 3\mathbf{i} - \mathbf{j} - \mathbf{k}.\]

1. Find \(\mathbf{a} \times \mathbf{b}\) and deduce the exact area of triangle \(OAB\).[3]
2. Hence, find the volume of the tetrahedron \(OABC\).
   [The volume of a tetrahedron is \(\frac{1}{3} \times \text{base area} \times \text{perpendicular height}\).][3]
3. Find the perpendicular distance from \(B\) to line \(OA\).[2]