Question

Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ be vectors of equal magnitude such that the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\alpha ,\overrightarrow{b}$ and $\overrightarrow{C}$ is $\beta$ and $\overrightarrow{c}$ and $\overrightarrow{a}$ 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. $-\frac{3}{2}$

Answer 1

Answer: (D)

$-\frac{3}{2}$

Let $|\overline{a}|=|\overline{b}|=|\overline{c}|=\lambda$
$\overline{a}.\overline{b}=|\overline{a}||\overline{b}|\cos \alpha$
$\therefore \overline{a}.\overline{b}={\lambda }^2\cos \alpha$

Similarly,$\overline{b}.\overline{c}={\lambda }^2\cos \beta$
$\overline{c}.\overline{a}={\lambda }^2\cos \gamma$

Now,$|\overline{a}+\overline{b}+\overline{c}{|}^2\ge 0$
$\therefore |\overline{a}{|}^2+|\overline{b}{|}^2+|\overline{c}{|}^2+2(\overline{a}.\overline{b}+\overline{b}.\overline{c}+\overline{c}.\overline{a})\ge 0$
$\Rightarrow 3(\lambda {)}^2+2(\lambda {)}^2(\cos \alpha +\cos \beta +\cos \gamma )\ge 0$
$\Rightarrow \cos \alpha +\cos \beta +\cos \gamma \ge \frac{-3}{2}$