Question

Let $\overline{a},\overline{b},\overline{c}$ be vectors equal magnitude such that the angle between $\overline{a}$ and $\overline{b}$ is $\alpha$, $\overline{b}$ and $\overline{c}$ is $\beta$, $\overline{c}$ and $\overline{a}$ 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)

$-\frac{3}{2}$

Given

$\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=\left|\overrightarrow{c}\right|=\lambda$

$\overrightarrow{a}.\overrightarrow{b}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|cos\alpha ={\lambda }^{2}cos\alpha$

$\overrightarrow{b}.\overrightarrow{c}=\left|\overrightarrow{b}\right|\left|\overrightarrow{c}\right|cos\beta ={\lambda }^{2}cos\beta$

$\overrightarrow{c}.\overrightarrow{a}=\left|\overrightarrow{c}\right|\left|\overrightarrow{a}\right|cos\gamma ={\lambda }^{2}cos\gamma$

we know $|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|\ge 0$

$\Rightarrow {\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2+{\left|\overrightarrow{c}\right|}^2+2(\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a})\ge 0$

$\Rightarrow 3{\lambda }^2+2({\lambda }^{2}cos\alpha +{\lambda }^{2}cos\beta +{\lambda }^{2}cos\gamma )\ge 0$

$\Rightarrow 3{\lambda }^2+2{\lambda }^2(cos\alpha +cos\beta +cos\gamma )\ge 0$

$\Rightarrow 3+2(cos\alpha +cos\beta +cos\gamma )\ge \frac{U}{{\lambda }^2}$

$\Rightarrow 2(cos\alpha +cos\beta +cos\gamma )\ge -3$

$\Rightarrow cos\alpha +cos\beta +cos\gamma \ge \frac{-3}{2}$