Given a regular hexagon $ABCDEF$, let $M$ be the midpoint of segment $DE$, $N$ be the midpoint of segment $AM$, and $P$ be the midpoint of segment $BC$. Express vector $\vec{NP}$ in terms of vectors $\vec{AB}$ and $\vec{AF}$.

1. Problem Description

Given a regular hexagon $ABCDEF$, let $M$ be the midpoint of segment $DE$, $N$ be the midpoint of segment $AM$, and $P$ be the midpoint of segment $BC$. Express vector $\vec{NP}$ in terms of vectors $\vec{AB}$ and $\vec{AF}$.

2. Solution Steps

Let $O$ be the center of the hexagon. We have $\vec{NP} = \vec{OP} - \vec{ON}$. Also, $\vec{OP} = \vec{OB} + \vec{BP} = \vec{OB} + \frac{1}{2}\vec{BC}$.

Since $ABCDEF$ is a regular hexagon, we can express vectors in terms of $\vec{AB}$ and $\vec{AF}$.

We have $\vec{OB} = \vec{OA} + \vec{AB}$. Also, $\vec{OA} = - \vec{OD}$.

Also, $\vec{OD} = \vec{OC} + \vec{CD} = \vec{OB} + \vec{CD}$.

Then, $\vec{OA} = - \vec{OD} = - \vec{OC} - \vec{CD} = - (\vec{OB} + \vec{CD})$.

Since $\vec{BC} = \vec{AO}$, $\vec{BC} = - \vec{OA}$.

Therefore, $\vec{OP} = \vec{OB} + \frac{1}{2}\vec{BC} = \vec{OB} - \frac{1}{2}\vec{OA}$.

We know $\vec{AM} = \vec{OM} - \vec{OA}$. Also $\vec{OM} = \frac{1}{2}(\vec{OD} + \vec{OE})$.

We have $\vec{OD} = \vec{AB} + \vec{AF}$ and $\vec{OE} = \vec{AF} + \vec{ED} = \vec{AF} - \vec{BA} = \vec{AF} - \vec{AB}$.

$\vec{OM} = \frac{1}{2}(\vec{AB} + \vec{AF} + \vec{AF} - \vec{AB}) = \frac{1}{2}(2\vec{AF}) = \vec{AF}$.

$\vec{OA} = \vec{AB} + \vec{BC} + \vec{CO} = -(\vec{OD} - \vec{AB} - \vec{OC}) = -(\vec{DC} + \vec{CO} - \vec{OB}) = \vec{FO} = -(\vec{BC} - \vec{AF} + \vec{FE} + \vec{ED}) = - (\vec{OC}) = \vec{AB}+\vec{AF}$

$\vec{OD} = \vec{AB} + \vec{AF}$. $\vec{OA} = - \vec{AB} - \vec{AF}$

$\vec{AM} = \vec{OM} - \vec{OA} = \vec{AF} - (- \vec{AB} - \vec{AF}) = \vec{AF} + \vec{AB} + \vec{AF} = \vec{AB} + 2\vec{AF}$.

$\vec{ON} = \vec{OA} + \vec{AN} = \vec{OA} + \frac{1}{2} \vec{AM} = - \vec{AB} - \vec{AF} + \frac{1}{2}(\vec{AB} + 2\vec{AF}) = - \vec{AB} - \vec{AF} + \frac{1}{2}\vec{AB} + \vec{AF} = - \frac{1}{2} \vec{AB}$.

$\vec{OP} = \vec{OB} + \frac{1}{2} \vec{BC} = \vec{AB} + \vec{OA} + \frac{1}{2} \vec{AO} = \vec{AB} - \vec{AB} - \vec{AF} - \frac{1}{2} \vec{OA} = \vec{AB} - (\vec{AB} + \vec{AF}) + \frac{1}{2} (\vec{AB} + \vec{AF}) = \vec{AF} + \frac{1}{2} \vec{AB}$

$\vec{OP} = \frac{1}{2}\vec{AB} - \vec{AF} = \vec{OB} - \frac{1}{2} \vec{AO} - \frac{1}{2} \vec{OC}= \frac{1}{2}\vec{AB} + \frac{1}{2} \vec{AF}$

$\vec{NP} = \vec{OP} - \vec{ON} = (\frac{1}{2}\vec{AB} + \frac{1}{2} \vec{AF}) - (- \frac{1}{2} \vec{AB}) = \frac{1}{2}\vec{AB} + \frac{1}{2} \vec{AF} + \frac{1}{2} \vec{AB} = \vec{AB} + \frac{1}{2}\vec{AF}$.

3. Final Answer

$\vec{NP} = \vec{AB} + \frac{1}{2}\vec{AF}$