Let $\vec a,\vec b,\vec c$ be vectors of equal magnitude such that the angle between $\vec a$ and $\vec b$ is $α$, $\vec b$ and $\vec c$ is $β$ and $\vec c$ and $\vec a$ is $γ$. Then, the minimum value of $\cos α + \cos β + \cos γ$, is
Answer & explanation
Correct answer: option 4
Let $|\vec a|=|\vec b|=|\vec c|=λ$
We have,
$\vec a.\vec b=|\vec a||\vec b|\cos α=λ^2\cos α$
$\vec b.\vec c=|\vec b||\vec c|\cos β=λ^2\cos β$
$\vec c.\vec a=|\vec c||\vec a|\cos γ=λ^2\cos γ$
Now,
$|\vec a+\vec b+\vec c|^2≥0$
$⇒|\vec a|^2+|\vec b|^2+|\vec c|^2+2(\vec a+\vec b+\vec b.\vec a+\vec c.\vec a)≥0$
$⇒3λ^2+2λ^2(\cos α + \cos β + \cos γ)≥0$
$⇒\cos α + \cos β + \cos γ≥-\frac{3}{2}$
Hence, the minimum value of $\cos α + \cos β + \cos γ$ is $-\frac{3}{2}$