Question #41622

The question tests the geometric interpretation of the vector triple product and the student's vigilance against the 'Zero-Factor Cancellation' trap.

When simplifying the relation $(\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} \cdot \vec{c})(\vec{b} \times \vec{c})$ via dot products, students arrive at an equation of the form $x \cdot k = x \cdot c \cdot V$, where $x = \vec{b} \cdot \vec{c}$.

The common mistake is to divide both sides by $x$, assuming it is non-zero, which yields a single, sub-optimal volume $V = -\sqrt{3}/6$.

By doing so, they miss the critical null-factor case where $x = 0$.

This case geometrically corresponds to $\vec{c}$ being perpendicular to both $\vec{a}$ and $\vec{b}$, maximizing the parallelepiped's volume to $1/2$.

The trap effectively separates students who mechanically cancel variables from those who rigorously analyze all algebraic branches in vector geometry.