Angle range when scalar product of vectors less or equal to zero

Question

Given two vectors $a$ and $b$, what is the possible angle between them when $ab \leq 0$?

Answer 1

Let $\vec{a}$ and $\vec{b}$ be two nonzero vectors. I'm going to denote their dot product by $\langle \vec{a},\vec{b} \rangle$. Now, we note that:

$$\langle \vec{a},\vec{b} \rangle = \|a\| \cdot \|b\| \cdot \cos(\theta)$$

where $\theta$ is the angle between them so that $\theta \in [0,\pi]$. Now, $\|a\| > 0$ and $\|b\| > 0$. So, all we need to do is to observe that:

$$\cos(\theta) \leq 0$$

and this is true when $\theta \in \left[\frac{\pi}{2},\pi \right]$. I hope that makes sense.