1. **Problem statement:** Given a regular hexagon centered at point $O$, with vectors $\overrightarrow{AB} = 3p + q$ and $\overrightarrow{BC} = 4p$, find: (a) $\overrightarrow{AO}$ in terms of $p$ and/or $q$. (b) $\overrightarrow{OB}$ in terms of $p$ and/or $q$. (c) $\overrightarrow{EB}$ in terms of $p$ and/or $q$. 2. **Key facts and formulas:** - The vector from point $X$ to point $Y$ is $\overrightarrow{XY} = \overrightarrow{OY} - \overrightarrow{OX}$. - Since $O$ is the center of the regular hexagon, the position vectors of vertices satisfy symmetry. - The hexagon has 6 vertices equally spaced around $O$. 3. **Find $\overrightarrow{OB}$:** - By definition, $\overrightarrow{OB}$ is the position vector of $B$ from $O$. - We know $\overrightarrow{AB} = 3p + q$. - Also, $\overrightarrow{OA} = ?$ (unknown yet). 4. **Express $\overrightarrow{AO}$:** - $\overrightarrow{AO} = \overrightarrow{O} - \overrightarrow{A} = -\overrightarrow{OA}$. 5. **Relate $\overrightarrow{OA}$ and $\overrightarrow{OB}$:** - $\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} = 3p + q$. - Rearranged: $\overrightarrow{OB} = \overrightarrow{OA} + 3p + q$. 6. **Express $\overrightarrow{BC}$:** - $\overrightarrow{BC} = \overrightarrow{OC} - \overrightarrow{OB} = 4p$. 7. **Use regular hexagon symmetry:** - The center $O$ is the midpoint of the hexagon. - The vector from $O$ to $A$ is opposite to the vector from $O$ to $D$ (the vertex opposite $A$). - The hexagon vertices are equally spaced by $60^\circ$. 8. **Find $\overrightarrow{OA}$ and $\overrightarrow{OB}$ in terms of $p$ and $q$:** - Since $\overrightarrow{AB} = 3p + q$ and $\overrightarrow{BC} = 4p$, and $B$ is adjacent to both $A$ and $C$, we can set $\overrightarrow{OB} = b$. - Then $\overrightarrow{OA} = b - (3p + q)$. - Also, $\overrightarrow{OC} = b + 4p$. 9. **Find $\overrightarrow{AO}$:** - $\overrightarrow{AO} = -\overrightarrow{OA} = -(b - (3p + q)) = -b + 3p + q$. 10. **Find $\overrightarrow{OB}$:** - $\overrightarrow{OB} = b$ (unknown vector). 11. **Find $\overrightarrow{EB}$:** - $\overrightarrow{EB} = \overrightarrow{OB} - \overrightarrow{OE}$. - Since $E$ is two vertices away from $B$ in the hexagon, $\overrightarrow{OE} = b + \overrightarrow{BC} + \overrightarrow{CD}$. - $\overrightarrow{BC} = 4p$. - $\overrightarrow{CD}$ is parallel and equal in magnitude to $\overrightarrow{AB} = 3p + q$ (since hexagon sides are equal). - So $\overrightarrow{OE} = b + 4p + (3p + q) = b + 7p + q$. - Therefore, $\overrightarrow{EB} = b - (b + 7p + q) = -7p - q$. 12. **Express $\overrightarrow{AO}$ and $\overrightarrow{OB}$ in simplest form:** - Since $O$ is the center, $\overrightarrow{OA} = -\overrightarrow{OC}$ and $\overrightarrow{OB} = -\overrightarrow{OE}$. - Using symmetry and the given vectors, the simplest expressions are: (a) $\overrightarrow{AO} = -3p - q$ (b) $\overrightarrow{OB} = 0$ (since $B$ is at the origin $O$ in this vector setup) (c) $\overrightarrow{EB} = -7p - q$ **Final answers:** $$\overrightarrow{AO} = -3p - q$$ $$\overrightarrow{OB} = 0$$ $$\overrightarrow{EB} = -7p - q$$