1. **Problem Statement:** Given vectors $\overrightarrow{AB} = 3p + q$ and $\overrightarrow{BC} = 4p$, and point $O$ inside a hexagon with vertices $A, B, C, D, E, F$, find the vectors $\overrightarrow{AO}$, $\overrightarrow{OB}$, and $\overrightarrow{EB}$ in terms of $p$ and $q$. 2. **Step a) Find $\overrightarrow{AO}$:** Assuming $O$ is the origin or center and the hexagon is regular, the vector from $A$ to $O$ is the negative of $\overrightarrow{OA}$. Because $\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} = 3p + q$, we can write $\overrightarrow{OA} + (3p + q) = \overrightarrow{OB}$. Assuming $O$ is the origin, $\overrightarrow{AO} = -\overrightarrow{OA}$. Since the exact relation is not given, the simplest form for $\overrightarrow{AO}$ is simply $$\overrightarrow{AO} = -\overrightarrow{OA}.$$ But we must deduce $\overrightarrow{AO}$ in terms of $p$ and $q$. Since $\overrightarrow{AB} = 3p + q$ and $\overrightarrow{BC} = 4p$, the vector $\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC} = 3p + q + 4p = 7p + q$. If $O$ is the midpoint of $AC$, then: $$\overrightarrow{AO} = \frac{1}{2} \overrightarrow{AC} = \frac{1}{2} (7p + q) = \frac{7}{2} p + \frac{1}{2} q,$$ which contains fractions, but the question wants answers without fractions. So we check another approach: It is common for the center $O$ of a regular hexagon to correspond to the vector sum of vectors from $O$ to vertices equals zero. Another way: If $O$ is the origin, then the position vectors are relative to $O$. Let $\overrightarrow{OA} = a$, $\overrightarrow{OB} = b$, $\overrightarrow{OC} = c$. Given $\overrightarrow{AB} = b - a = 3p + q$, so $b = a + 3p + q$. Given $\overrightarrow{BC} = c - b = 4p$, so $c = b + 4p = a + 3p + q + 4p = a +7p + q$. Because $O$ is at origin: $\overrightarrow{OA} = a$, $\overrightarrow{OB} = b$, $\overrightarrow{OC} = c$. Hence: - $\overrightarrow{AO} = -a$ - $\overrightarrow{OB} = b = a + 3p + q$ We don't know $a$, but we can express $AO$ and $OB$ in terms of $a$, $p$, $q$. 3. **Step b) Find $\overrightarrow{OB}$:** From above, $\overrightarrow{OB} = a + 3p + q$. 4. **Step c) Find $\overrightarrow{EB}$:** Assuming the hexagon is regular with vertices labeled $A$ through $F$ clockwise, and vectors are consistent, the vector $\overrightarrow{EB} = \overrightarrow{EB} = ?$. Since E is two vertices after C (E is at vertex 5), and knowing no other vectors, Calculate $\overrightarrow{EB} = \overrightarrow{OB} - \overrightarrow{OE}$. Assuming $O$ is origin, say: $\overrightarrow{OE} = e$, $\overrightarrow{OB} = b$. We can try expressing $e$ in terms of $a$, $p$, and $q$ based on regular hexagon rotational symmetry. For a regular hexagon, displacement vectors between consecutive vertices correspond to the given vectors. If $\overrightarrow{AB} = 3p + q$, and $\overrightarrow{BC} = 4p$, then using symmetry: $\overrightarrow{CD} = 3p + q$ (rotated), $\overrightarrow{DE} = 4p$ (rotated), so total from B to E is $\overrightarrow{BE} = \overrightarrow{BC} + \overrightarrow{CD} + \overrightarrow{DE} = 4p + (3p + q) + 4p = 11p + q$. Therefore, $$\overrightarrow{EB} = - \overrightarrow{BE} = - (11p + q) = -11p - q.$$ 5. **Final answers:** Since no fixed origin position vector $a$ is given, express answers in simplest form assuming $\overrightarrow{OA} = a$: - a) $\overrightarrow{AO} = -a$ - b) $\overrightarrow{OB} = a + 3p + q$ - c) $\overrightarrow{EB} = -11p - q$ These are the simplest forms without fractions and using given vectors.