Question

Let \( \vec{a}, \vec{b}, \vec{c} \) be vectors of equal magnitude such that the angle between \( \vec{a} \) and \( \vec{b} \) is \( \alpha \), the angle between \( \vec{b} \) and \( \vec{c} \) is \( \beta \), and the angle between \( \vec{c} \) and \( \vec{a} \) is \( \gamma \). Then the minimum value of \( \cos \alpha + \cos \beta + \cos \gamma \) is:

• A. \( \frac{1}{2} \)
• B. \( \frac{3}{2} \)
• C. \( \frac{1}{3} \)
• D. \( -\frac{3}{2} \)

Hint:

For problems involving minimizing the sum of cosines of angles between vectors, consider using symmetry, especially when vectors are of equal magnitude. The minimum value is often achieved when the vectors are symmetrically arranged.