Given a regular hexagon $ABCDEF$, and the vectors $\vec{AB} = \vec{m}$ and $\vec{BC} = \vec{n}$, express the vectors $\vec{BE}$, $\vec{DF}$, and $\vec{CD}$ in terms of $\vec{m}$ and $\vec{n}$.

1. Problem Description

Given a regular hexagon $ABCDEF$, and the vectors $\vec{AB} = \vec{m}$ and $\vec{BC} = \vec{n}$, express the vectors $\vec{BE}$, $\vec{DF}$, and $\vec{CD}$ in terms of $\vec{m}$ and $\vec{n}$.

2. Solution Steps

In a regular hexagon $ABCDEF$, all sides have the same length, and all interior angles are equal to $120^{\circ}$. Also opposite sides are parallel and equal in length.

First, let us express $\vec{BE}$ in terms of $\vec{m}$ and $\vec{n}$. We can write $\vec{BE} = \vec{BC} + \vec{CD} + \vec{DE}$. Since $ABCDEF$ is a regular hexagon, $\vec{CD} = \vec{BA} = -\vec{AB} = -\vec{m}$. Also, $\vec{DE} = \vec{BC} = \vec{n}$. Therefore,

$$\vec{BE} = \vec{n} - \vec{m} + \vec{n} = 2\vec{n} - \vec{m}.$$

Next, let's find $\vec{DF}$. We have $\vec{DF} = \vec{DE} + \vec{EF}$. Since $\vec{DE} = \vec{BC} = \vec{n}$ and $\vec{EF} = \vec{BA} = -\vec{AB} = -\vec{m}$, then $\vec{DF} = \vec{n} - \vec{m}$.

Finally, $\vec{CD}$ is opposite and equal in length to $\vec{AB}$ but has the opposite direction. Therefore $\vec{CD} = -\vec{AB} = -\vec{m}$.

3. Final Answer

$\vec{BE} = 2\vec{n} - \vec{m}$

$\vec{DF} = \vec{n} - \vec{m}$

$\vec{CD} = -\vec{m}$