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.

Answer 1

The Correct Option is D

We are given three vectors \( \vec{a}, \vec{b}, \vec{c} \) of equal magnitude. The angles between the vectors are \( \alpha, \beta, \gamma \), respectively. We are asked to find the minimum value of: \[ \cos \alpha + \cos \beta + \cos \gamma \]
Step 1: Symmetry of the vectors
Since the vectors \( \vec{a}, \vec{b}, \vec{c} \) are of equal magnitude, the sum \( \cos \alpha + \cos \beta + \cos \gamma \) is minimized when the vectors are symmetrically arranged. The simplest configuration is when all the angles between the vectors are equal. This occurs when \( \alpha = \beta = \gamma = 120^\circ \), forming an equilateral triangle in three-dimensional space.
Step 2: Calculate the cosine values
For \( \alpha = \beta = \gamma = 120^\circ \), we know: \[ \cos 120^\circ = -\frac{1}{2} \] Thus, the sum becomes: \[ \cos \alpha + \cos \beta + \cos \gamma = 3 \times \left( -\frac{1}{2} \right) = -\frac{3}{2} \]
Step 3: Use of vectors
Given that this problem asks for the minimum value of the sum of cosines, the most symmetrical arrangement of vectors results in this value when the angle between any two vectors is \( 120^\circ \). This configuration minimizes the sum of cosines.
Step 4: Conclusion
Therefore, the minimum value of \( \cos \alpha + \cos \beta + \cos \gamma \) is: \[ \boxed{-\frac{3}{2}} \]
Final Answer: \( \boxed{(D) -\frac{3}{2}} \)