Mathematics

Problem

The volume of the parallelopiped whose coterminous edges are represented by the vectors

2\mathbf{b} \times \mathbf{c}

3\mathbf{c} \times \mathbf{a}

and

4\mathbf{a} \times \mathbf{b}

where

\mathbf{a} = (1 + \sin \theta)\hat{\mathbf{i}} + \cos \theta \hat{\mathbf{j}} + \sin 2\theta \hat{\mathbf{k}}

\mathbf{b} = \sin\left(\theta + \frac{2\pi}{3}\right)\hat{\mathbf{i}} + \cos\left(\theta + \frac{2\pi}{3}\right)\hat{\mathbf{j}} + \sin\left(2\theta + \frac{4\pi}{3}\right)\hat{\mathbf{k}},

\mathbf{c} = \sin\left(\theta - \frac{2\pi}{3}\right)\hat{\mathbf{i}} + \cos\left(\theta - \frac{2\pi}{3}\right)\hat{\mathbf{j}} + \sin\left(2\theta - \frac{4\pi}{3}\right)\hat{\mathbf{k}} \text{ is 18}

cubic units, then the value of

\theta

in the interval

\left[0, \frac{\pi}{2}\right]

, is/are a.

\frac{\pi}{9}

\frac{2\pi}{9}

\frac{\pi}{3}

\frac{4\pi}{9}