Let $\vec a,\vec b,\vec c$ be three unit vectors such that angle between $\vec a$ and $\vec b$ is $α$, $\vec b$ and $\vec c$ is $β$ and $\vec c$ and $\vec a$ is $γ$. If $|\vec a+\vec b+\vec c|=2$, then $\cos α + \cos β + \cos γ =$
Answer & explanation
Correct answer: option 4
We have,
$\vec a.\vec b = \cos α, \vec b. \vec c = \cos β$ and $\vec c. \vec a = \cos γ$
Now,
$|\vec a+\vec b+\vec c|=2$
$⇒|\vec a+\vec b+\vec c|^2=4$
$⇒|\vec a|+|\vec b|^2+|\vec c|^2+2(\vec a.\vec b+\vec b. \vec c+\vec c. \vec a)=4$
$⇒1+1+1+2(\cos α+\cos β+\cos γ)=4$
$⇒\cos α+\cos β+\cos γ=\frac{1}{2}$