1. **Problem statement:** We have a regular hexagon centered at the origin O. Given: - Side AB = $3p + q$ - Side BC = $4p$ We need to find: (a) Position vector of D in terms of $p$ and/or $q$. (b) Position vector of B in terms of $p$ and/or $q$. (c) Vector EB in terms of $p$ and/or $q$. 2. **Important properties and formulas:** - In a regular hexagon, all sides are equal in length. - The position vectors of vertices can be expressed relative to O. - The hexagon vertices are equally spaced at 60° angles. - Vector addition and subtraction rules apply. 3. **Step (a): Position vector of D** - Since AB = $3p + q$ and BC = $4p$, and the hexagon is regular, all sides equal $3p + q = 4p$. - This implies $3p + q = 4p \Rightarrow q = p$. - The vertices are arranged at 60° increments around O. - Let’s assign vectors: - $\vec{A} = 3p + q$ - $\vec{B} = ?$ - $\vec{C} = 4p$ - Since $q = p$, $3p + q = 4p$. - The vector from O to D is opposite to B (since hexagon is symmetric), so: $$\vec{D} = -\vec{B}$$ - We find $\vec{B}$ first. 4. **Step (b): Position vector of B** - Given $\vec{AB} = 3p + q = 4p$ (since $q=p$), and $\vec{BC} = 4p$. - Since B is at upper-left, and A is at left, the vector $\vec{B}$ can be expressed as: $$\vec{B} = \vec{A} + \vec{AB}$$ - But $\vec{A} = 3p + q = 4p$, so: $$\vec{B} = 4p + 4p = 8p$$ - However, this contradicts the hexagon side length equality, so we consider the directions. - Using the hexagon geometry, the position vector of B is: $$\vec{B} = 3p + q$$ 5. **Step (c): Vector EB** - E is the vertex opposite to B. - Since O is center, $\vec{E} = -\vec{B}$. - Vector $\vec{EB} = \vec{B} - \vec{E} = \vec{B} - (-\vec{B}) = 2\vec{B}$. - Using $\vec{B} = 3p + q$, we get: $$\vec{EB} = 2(3p + q) = 6p + 2q$$ 6. **Final answers:** - (a) $\boxed{\vec{D} = - (3p + q)}$ - (b) $\boxed{\vec{B} = 3p + q}$ - (c) $\boxed{\vec{EB} = 6p + 2q}$ These are the simplest forms in terms of $p$ and $q$.