Question #41388

Let $\vec{u}, \vec{v}, \vec{w}$ be three unit vectors such that the angle between any two of them is $\frac{\pi}{3}$. Define three new vectors using the vector triple product: $\vec{a} = \vec{u} \times (\vec{v} \times \vec{w}), \quad \vec{b} = \vec{v} \times (\vec{w} \times \vec{u}), \quad \vec{c} = \vec{w} \times (\vec{u} \times \vec{v})$ A variable parallelepiped is formed by the coterminous edges $\vec{A} = \vec{u} + x\vec{a}$, $\vec{B} = \vec{v} + x\vec{b}$, and $\vec{C} = \vec{w} + x\vec{c}$, where $x$ is a scalar parameter. Let $\Delta V$ be the exact change in the volume of this parallelepiped as $x$ changes from $0$ to a small value $\epsilon > 0$. If $dV$ is the estimated change in volume using the first-order differential approximation at $x=0$, find the value of $\Delta V - dV$.