Given three vectors \( \vec{a}, \vec{b}, \vec{c } \) of equal magnitude, with angles \( \alpha, \beta, \gamma \) between them, find the minimum value of:
\[\n\cos \alpha + \cos \beta + \cos \gamma\n\]
Solution:
The sum \( \cos \alpha + \cos \beta + \cos \gamma \) is minimized when the vectors are symmetrically arranged. The simplest symmetrical configuration is when \( \alpha = \beta = \gamma = 120^\circ \).
Step 1: Angle calculation
For \( \alpha = \beta = \gamma = 120^\circ \):
\[\n\cos 120^\circ = -\frac{1}{2}\n\]
Step 2: Sum Calculation
\[\n\cos \alpha + \cos \beta + \cos \gamma = 3 \times \left( -\frac{1}{2} \right) = -\frac{3}{2}\n\]
Step 3: Justification
The most symmetrical arrangement with \( 120^\circ \) angles minimizes the sum of cosines.
Step 4: Answer
The minimum value is:
\[\n\boxed{-\frac{3}{2}}\n\]
Final Answer: \( \boxed{(D) -\frac{3}{2}} \)