Problem

Figure 1

The points A, B and C have position vectors $a$, $b$ and $c$ respectively, relative to a fixed origin O, as shown in Figure 1, where

$$a = 2\mathbf{i} + \mathbf{k} \qquad b = \mathbf{i} + 2\mathbf{j} - 2\mathbf{k} \qquad c = 3\mathbf{i} + \mathbf{j} + \mathbf{k}$$

Determine

(a) $b \times c$

(b) $a \cdot (b \times c)$

(c) the volume of the tetrahedron $OABC$

(d) the shortest distance from A to the plane containing O, B and C, giving the answer as a simplified surd.