If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a} \cdot \vec{b} = 0 = \vec{a} \cdot \vec{c}$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{3}$. Then the value of $|\vec{a} \times \vec{b} - \vec{a} \times \vec{c}|$ is ______.
Step-by-Step Solution
Key Concept: When two vectors are perpendicular, the magnitude of their cross product equals the product of their magnitudes.
Given $\vec{a} \cdot \vec{b} = 0 = \vec{a} \cdot \vec{c}$ and $|\vec{a} \times (\vec{b} - \vec{c})| = |\vec{a}| |\vec{b} - \vec{c}|$, since $\vec{a}$ is perpendicular to $\vec{b} - \vec{c}$ as $\vec{a} \cdot (\vec{b} - \vec{c}) = 0$, we have $|\vec{a}| = 1$ and $|\vec{b} - \vec{c}| = \sqrt{|\vec{b}|^2 + |\vec{c}|^2 - 2\vec{b} \cdot \vec{c}}$.
Correct Answer: Looking at this problem step-by-step:
**Given:**
- $\vec{a}, \vec{b}, \vec{c}$ are unit vectors: $|\vec{a}| = |\vec{b}| = |\vec{c}| = 1$
- $\vec{a} \cdot \vec{b}