1. **Problem statement:** We are given vectors \(\overrightarrow{OA} = a\), \(\overrightarrow{OC} = c\), and \(\overrightarrow{AB} = b\). Point \(P\) lies on line \(AB\) such that \(AP : PB = 2 : 1\), and \(Q\) is the midpoint of \(BC\). We need to find: a. \(\overrightarrow{CB}\) b. The position vector of \(Q\) c. \(\overrightarrow{PQ}\) --- 2. **Recall vector rules:** - Vector addition: \(\overrightarrow{XY} = \overrightarrow{OY} - \overrightarrow{OX}\) - Midpoint of two points \(M\) between \(X\) and \(Y\): \(\overrightarrow{OM} = \frac{\overrightarrow{OX} + \overrightarrow{OY}}{2}\) - Dividing a segment in ratio \(m:n\): If \(P\) divides \(AB\) in ratio \(m:n\), then \(\overrightarrow{OP} = \frac{n\overrightarrow{OA} + m\overrightarrow{OB}}{m+n}\) --- 3. **Find \(\overrightarrow{CB}\):** \[\overrightarrow{CB} = \overrightarrow{OB} - \overrightarrow{OC}\] Since \(\overrightarrow{OB} = \overrightarrow{OA} + \overrightarrow{AB} = a + b\), \[\overrightarrow{CB} = (a + b) - c = a + b - c\] --- 4. **Find position vector of \(Q\), midpoint of \(BC\):** \[\overrightarrow{OQ} = \frac{\overrightarrow{OB} + \overrightarrow{OC}}{2} = \frac{(a + b) + c}{2} = \frac{a + b + c}{2}\] --- 5. **Find \(\overrightarrow{PQ}\):** First find \(\overrightarrow{OP}\). Since \(P\) divides \(AB\) in ratio \(2:1\), \[\overrightarrow{OP} = \frac{1 \cdot \overrightarrow{OA} + 2 \cdot \overrightarrow{OB}}{2 + 1} = \frac{a + 2(a + b)}{3} = \frac{a + 2a + 2b}{3} = \frac{3a + 2b}{3} = a + \frac{2}{3}b\] Then, \[\overrightarrow{PQ} = \overrightarrow{OQ} - \overrightarrow{OP} = \frac{a + b + c}{2} - \left(a + \frac{2}{3}b\right) = \frac{a + b + c}{2} - a - \frac{2}{3}b\] Simplify: \[= \frac{a + b + c}{2} - \frac{2a}{2} - \frac{4b}{6} = \frac{a + b + c - 2a}{2} - \frac{4b}{6} = \frac{-a + b + c}{2} - \frac{4b}{6}\] Convert \(\frac{4b}{6} = \frac{2b}{3}\) to denominator 6: \[\frac{-a + b + c}{2} = \frac{-3a + 3b + 3c}{6}\] So, \[\overrightarrow{PQ} = \frac{-3a + 3b + 3c}{6} - \frac{4b}{6} = \frac{-3a + 3b + 3c - 4b}{6} = \frac{-3a - b + 3c}{6}\] --- **Final answers:** \[\boxed{\overrightarrow{CB} = a + b - c}\] \[\boxed{\overrightarrow{OQ} = \frac{a + b + c}{2}}\] \[\boxed{\overrightarrow{PQ} = \frac{-3a - b + 3c}{6}}\]