Cross Product of Three Coplanar Unit Vectors Summing to Zero

Given $\vec{a} + \vec{b} + \vec{c} = \vec{0}$.

1. Taking the cross product with $\vec{a}$:
   $\vec{a} \times (\vec{a} + \vec{b} + \vec{c}) = \vec{0} \implies \vec{0} + \vec{a} \times \vec{b} + \vec{a} \times \vec{c} = \vec{0} \implies \vec{a} \times \vec{b} = \vec{c} \times \vec{a}$

2. Taking the cross product with $\vec{b}$:
   $\vec{b} \times (\vec{a} + \vec{b} + \vec{c}) = \vec{0} \implies \vec{b} \times \vec{a} + \vec{0} + \vec{b} \times \vec{c} = \vec{0} \implies \vec{b} \times \vec{c} = \vec{a} \times \vec{b}$

Thus, $\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a}$.

1. Since $|\vec{a}| = |\vec{b}| = |\vec{c}| = 1$ and $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, the vectors form an equilateral triangle in a plane with angle $120^\circ$ between any two vectors. Since they are non-collinear:
   $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin 120^\circ = 1 \cdot 1 \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \neq 0$

Hence, $\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a} \neq \vec{0}$.

Correct Answer: B