$(\vec{a} \times \vec{b}) \times \vec{c} = \vec{a} \times (\vec{b} \times \vec{c})$ is $\vec{a}$ and $\vec{c}$ are:
Step-by-Step Solution
Key Concept: The vector triple product formula combined with equality of two expressions reveals that $\vec{a}$ and $\vec{c}$ must be collinear.
Using the vector triple product formula: $(\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{b} \cdot \vec{c})\vec{a}$ and $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$. For these to be equal: $(\vec{a} \cdot \vec{c})\vec{b} - (\vec{b} \cdot \vec{c})\vec{a} = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$. This simplifies to $(\vec{b} \cdot \vec{c})\vec{a} = (\vec{a} \cdot \vec{b})\vec{c}$. This equation holds when $\vec{a}$ and $\vec{c}$ are collinear (scalar multiples of each other), i.e., $\vec{a} = k\vec{c}$ for some scalar $k$.
Correct Answer: 2