1. **Problem statement:** Given a triangle ABC with vectors $\vec{a} = \overrightarrow{AB}$ and $\vec{b} = \overrightarrow{AC}$, find the vectors represented by: (i) $\overrightarrow{BC}$ (ii) $\overrightarrow{CB}$ (iii) $\overrightarrow{AD}$, where D is the midpoint of BC. 2. **Recall vector addition and subtraction rules:** - $\overrightarrow{XY} = \overrightarrow{OY} - \overrightarrow{OX}$ for points X, Y, and origin O. - The midpoint D of BC satisfies $\overrightarrow{OD} = \frac{\overrightarrow{OB} + \overrightarrow{OC}}{2}$. 3. **Express vectors in terms of $\vec{a}$ and $\vec{b}$:** - Since $\vec{a} = \overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}$, and $\vec{b} = \overrightarrow{AC} = \overrightarrow{OC} - \overrightarrow{OA}$, we can write: 4. **Find $\overrightarrow{BC}$:** $$\overrightarrow{BC} = \overrightarrow{OC} - \overrightarrow{OB} = (\overrightarrow{OA} + \vec{b}) - (\overrightarrow{OA} + \vec{a}) = \vec{b} - \vec{a}$$ 5. **Find $\overrightarrow{CB}$:** $$\overrightarrow{CB} = \overrightarrow{OB} - \overrightarrow{OC} = (\overrightarrow{OA} + \vec{a}) - (\overrightarrow{OA} + \vec{b}) = \vec{a} - \vec{b}$$ 6. **Find $\overrightarrow{AD}$:** - Since D is midpoint of BC, $$\overrightarrow{OD} = \frac{\overrightarrow{OB} + \overrightarrow{OC}}{2} = \frac{(\overrightarrow{OA} + \vec{a}) + (\overrightarrow{OA} + \vec{b})}{2} = \overrightarrow{OA} + \frac{\vec{a} + \vec{b}}{2}$$ - Therefore, $$\overrightarrow{AD} = \overrightarrow{OD} - \overrightarrow{OA} = \frac{\vec{a} + \vec{b}}{2}$$ **Final answers:** (i) $\overrightarrow{BC} = \vec{b} - \vec{a}$ (ii) $\overrightarrow{CB} = \vec{a} - \vec{b}$ (iii) $\overrightarrow{AD} = \frac{\vec{a} + \vec{b}}{2}$