Step-by-Step Solution

The dot product of two vectors $\vec{a}$ and $\vec{b}$ is given by:

$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos{\theta}$

We are given that $\vec{a} \cdot \vec{b} \ge 0$. Since $|\vec{a}|$ and $|\vec{b}|$ are magnitudes, they are always non-negative. Therefore, for the dot product to be non-negative, we must have:

$\cos{\theta} \ge 0$

The cosine function is non-negative in the first and fourth quadrants. However, since $\theta$ is the angle between two vectors, it must lie between $0$ and $\pi$ (inclusive), i.e., $0 \le \theta \le \pi$.

In the interval $[0, \pi]$, $\cos{\theta} \ge 0$ when $0 \le \theta \le \frac{\pi}{2}$.