Given a regular hexagon $ABCDEF$, and vectors $\vec{AB} = \vec{p}$ and $\vec{BC} = \vec{q}$, find the vectors $\vec{CD}$, $\vec{DE}$, $\vec{EF}$, $\vec{FA}$, $\vec{AD}$, $\vec{EA}$, and $\vec{AC}$ in terms of $\vec{p}$ and $\vec{q}$.

1. Problem Description

Given a regular hexagon $ABCDEF$, and vectors $\vec{AB} = \vec{p}$ and $\vec{BC} = \vec{q}$, find the vectors $\vec{CD}$, $\vec{DE}$, $\vec{EF}$, $\vec{FA}$, $\vec{AD}$, $\vec{EA}$, and $\vec{AC}$ in terms of $\vec{p}$ and $\vec{q}$.

2. Solution Steps

Since $ABCDEF$ is a regular hexagon, all sides have equal length, and each interior angle is 120 degrees.

$\vec{CD}$ : Since $ABCDEF$ is a regular hexagon, $\vec{CD}$ has the same length as $\vec{AB}$. The angle between $\vec{BC}$ and $\vec{CD}$ is also 120 degrees. Therefore, $\vec{CD} = -\vec{p}$.

$\vec{DE}$ : Since $ABCDEF$ is a regular hexagon, $\vec{DE}$ has the same length as $\vec{BC}$ but the opposite direction. So, $\vec{DE} = -\vec{q}$.

$\vec{EF}$ : $\vec{EF}$ has the same length as $\vec{AB}$, and has the same direction as $\vec{BA}$. So $\vec{EF} = -\vec{p}$.

$\vec{FA}$ : $\vec{FA}$ has the same length as $\vec{BC}$, and has the same direction as $\vec{CB}$. So $\vec{FA} = -\vec{q}$.

$\vec{AD}$ : $\vec{AD} = \vec{AB} + \vec{BC} + \vec{CD} = \vec{p} + \vec{q} + (-\vec{p}) = \vec{q} + \vec{p} - \vec{p} + \vec{BC} = \vec{AD} = \vec{BC}+\vec{EF}= \vec{AB}+\vec{CD}+2*\vec{BC} = \vec{CD}-\vec{p}=\vec{AB}=2\vec{BC}$.

Since $\vec{AD}$ is parallel to $\vec{BC}$, and has twice the length, $\vec{AD} = 2\vec{BC} = 2\vec{q}$.

$\vec{EA}$ : $\vec{EA} = -\vec{AE}$. We have $\vec{AE} = \vec{AD} + \vec{DE} = 2\vec{q} - \vec{q} = \vec{q}$. So $\vec{EA} = -\vec{q}$. Also, $\vec{EA} = \vec{FA} - \vec{FE} = -\vec{q} + \vec{p}$. So $\vec{EA} = \vec{AB}+\vec{FA}= \vec{CD}$

$\vec{AC}$ : $\vec{AC} = \vec{AB} + \vec{BC} = \vec{p} + \vec{q}$.

3. Final Answer

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

$\vec{DE} = -\vec{q}$

$\vec{EF} = -\vec{p}$

$\vec{FA} = -\vec{q}$

$\vec{AD} = 2\vec{q}$

$\vec{EA} = \vec{p}-\vec{q}$

$\vec{AC} = \vec{p} + \vec{q}$