Finding angle between two vectors

Rearranging the dot-product definition, the angle $\theta$ ($0\le\theta\le\pi$) between two non-zero vectors

$\bar a,\bar b$ is

Since cosine is positive for $0\le\theta<\pi/2$, zero at $\theta=\pi/2$, and negative for $\pi/2<\theta\le\pi$,

the sign of $\bar a\cdot\bar b$ alone already tells whether the angle between two vectors is acute

($\bar a\cdot\bar b>0$), right ($\bar a\cdot\bar b=0$), or obtuse ($\bar a\cdot\bar b<0$) -- without ever

computing $\theta$ itself. This sign test is exactly how "obtuse angle" conditions in the exercises are turned …