If $|\vec{a}| = |\vec{b}| = |\vec{c}| = 2$ and $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a} = -1$, then $|\vec{a} \times \vec{b} \times \vec{c} \times \vec{a}|$ is_______.
Step-by-Step Solution
Key Concept: The scalar triple product of cross products equals the square of the original triple product.
For the scalar triple product $[\vec{a} \times \vec{b} \ \vec{b} \times \vec{c} \ \vec{c} \times \vec{a}] = [\vec{a} \ \vec{b} \ \vec{c}]^2$, we evaluate the determinant by expressing each cross product in terms of the original vectors. The result is shown as a $3 \times 3$ determinant equal to $16$.
Correct Answer: I need to find $|\vec{a} \times \vec{b} \times \vec{c} \times \vec{a}|$.
First, let me clarify the notation. The expression $\vec{a} \times \vec{b} \times \vec{c} \times \vec{a}$ likely means $(\vec{a