The problem asks us to simplify the expression $\vec{AB} - \vec{AC} + \vec{BD} - \vec{CD}$.

1. Problem Description

The problem asks us to simplify the expression $\vec{AB} - \vec{AC} + \vec{BD} - \vec{CD}$.

2. Solution Steps

We can rewrite the expression using the properties of vector addition and subtraction.

$\vec{AB} - \vec{AC} + \vec{BD} - \vec{CD}$

First, we use the property that $\vec{AB} - \vec{AC} = \vec{CB}$.

So, $\vec{AB} - \vec{AC} = \vec{CB}$.

Then we can rewrite $\vec{BD} - \vec{CD}$ as $\vec{BD} + \vec{DC} = \vec{BC}$.

$\vec{AB} - \vec{AC} + \vec{BD} - \vec{CD} = \vec{CB} + \vec{BC}$

Since $\vec{CB} = -\vec{BC}$,

$\vec{CB} + \vec{BC} = -\vec{BC} + \vec{BC} = \vec{0}$

3. Final Answer

The final answer is D. $\vec{0}$