Vector Algebra
Volume of Parallelepiped
Grade 12
Question:
<p>The volume of the parallelepiped 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 $\mathbf{a} = (1 + \sin \theta)\mathbf{i} + \cos \theta \mathbf{j} + \sin 2\theta \mathbf{k}$, $\mathbf{b} = \sin\left(\theta + \frac{2\pi}{3}\right)\mathbf{i} + \cos\left(\theta + \frac{2\pi}{3}\right)\mathbf{j} + \sin\left(2\theta + \frac{4\pi}{3}\right)\mathbf{k}$, $\mathbf{c} = \sin\left(\theta - \frac{2\pi}{3}\right)\mathbf{i} + \cos\left(\theta - \frac{2\pi}{3}\right)\mathbf{j} + \sin\left(2\theta - \frac{4\pi}{3}\right)\mathbf{k}$ is 18 cubic units, then the value of $\theta$ in the interval $\left(0, \frac{\pi}{2}\right)$ is:</p>
<p>(a) $\frac{\pi}{9}$</p>
<p>(b) $\frac{2\pi}{9}$</p>
<p>(c) $\frac{\pi}{3}$</p>
<p>(d) $\frac{4\pi}{9}$</p>
Step-by-Step Solution
Key Concept: The volume of a parallelepiped with coterminous edges is the absolute value of the scalar triple product. Use the property that the scalar triple product of vectors obtained from cyclic products can be simplified.
Solution: Volume $= |[2\mathbf{b} \times \mathbf{c}, 3\mathbf{c} \times \mathbf{a}, 4\mathbf{a} \times \mathbf{b}]| = 18$ $\Rightarrow 24[\mathbf{a}, \mathbf{b}, \mathbf{c}]^2 = 18$ $\Rightarrow |[\mathbf{a}, \mathbf{b}, \mathbf{c}]| = \frac{3}{2}$ Computing the scalar triple product using the given vectors and solving for $\theta$.
Correct Answer: a, b, d