Question

Let \( \vec{a}, \vec{b}, \vec{c} \) be vectors of equal magnitude such that the angle between \( \vec{a} \) and \( \vec{b} \) is \( \alpha \), between \( \vec{b} \) and \( \vec{c} \) is \( \beta \), and 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. \( -\frac{1}{2} \)
• C. \( \frac{3}{2} \)
• D. \( -\frac{3}{2} \)

Hint:

For vector angle sum problems: \begin{itemize} \item Use \( |\vec{a}+\vec{b}+\vec{c}|^2 \ge 0 \). \item Convert dot products to cosines. \item Equality occurs when vector sum is zero. \end{itemize}

Answer 1

The Correct Option is D