Vector Db D3B040

1. **State the problem:** We have a parallelogram OABC with vectors $\overrightarrow{OA} = 14a$ and $\overrightarrow{OB} = 7b$. Point D lies on segment AC such that the ratio $AD : DC = 2 : 5$. We need to find the vector $\overrightarrow{DB}$ in terms of $a$ and $b$. 2. **Express vector $\overrightarrow{AC}$:** Since $\overrightarrow{AC} = \overrightarrow{OC} - \overrightarrow{OA}$ and $\overrightarrow{OC} = \overrightarrow{OA} + \overrightarrow{OB}$ (because OABC is a parallelogram), we have: $$\overrightarrow{AC} = (\overrightarrow{OA} + \overrightarrow{OB}) - \overrightarrow{OA} = \overrightarrow{OB} = 7b$$ 3. **Find vector $\overrightarrow{AD}$:** Since $D$ divides $AC$ in ratio $2:5$, the length of $AD$ is $\frac{2}{2+5} = \frac{2}{7}$ of $AC$. Therefore, $$\overrightarrow{AD} = \frac{2}{7} \overrightarrow{AC} = \frac{2}{7} \times 7b = 2b$$ 4. **Find position vector of D:** $$\overrightarrow{OD} = \overrightarrow{OA} + \overrightarrow{AD} = 14a + 2b$$ 5. **Find vector $\overrightarrow{DB}$:** $$\overrightarrow{DB} = \overrightarrow{OB} - \overrightarrow{OD} = 7b - (14a + 2b) = -14a + (7b - 2b) = -14a + 5b$$ **Final answer:** $$\boxed{\overrightarrow{DB} = -14a + 5b}$$