The problem asks to calculate the dot product of two vectors $\vec{a}$ and $\vec{b}$, given that the angle between them is 180 degrees, $|\vec{a}| = 2$ and $|\vec{b}| = 5$.

1. Problem Description

The problem asks to calculate the dot product of two vectors $\vec{a}$ and $\vec{b}$, given that the angle between them is 180 degrees, $|\vec{a}| = 2$ and $|\vec{b}| = 5$.

2. Solution Steps

We use the formula for the dot product of two vectors:

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

where $\theta$ is the angle between the two vectors. In this case, $\theta = 180^{\circ}$. We know that $\cos{180^{\circ}} = -1$.

Thus,

$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos{180^{\circ}} = (2)(5)(-1) = -10$

3. Final Answer

$\vec{a} \cdot \vec{b} = -10$