Question #41616

This question tests the properties of reciprocal vector systems and the scalar triple product, specifically targeting the 'Square Root and Modulus Logic' trap.

Students often mechanically assume that $\sqrt{x^2} = x$, forgetting the absolute value definition $\sqrt{x^2} = |x|$.

When evaluating the term $\sqrt{[\vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}]} = \sqrt{[\vec{a} \; \vec{b} \; \vec{c}]^2}$, they incorrectly simplify it to $[\vec{a} \; \vec{b} \; \vec{c}]$, leading to a final answer of $0$.

The negative determinant of the coefficient matrix ensures $[\vec{a} \; \vec{b} \; \vec{c}]$ is negative, forcing the distinction between $x$ and $|x|$ to be the deciding factor in reaching the correct answer.