Practicing Success
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 |
$\frac{1}{2}$ $-\frac{1}{2}$ $\frac{3}{2}$ $-\frac{3}{2}$ |
$-\frac{3}{2}$ |
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 $-\frac{3}{2}$ |