1. **State the problem:** We have a parallelogram OXYZ with vectors \(\overrightarrow{OX} = a\) and \(\overrightarrow{OY} = b\). Points \(P\) and \(R\) divide \(OX\) and \(OY\) in ratios 1:2 and 1:3 respectively. We need to find the ratio \(ZP : ZR\). 2. **Recall properties of parallelograms:** The vector \(\overrightarrow{OZ} = \overrightarrow{OX} + \overrightarrow{OY} = a + b\). 3. **Find position vectors of points P and R:** - Since \(OP : PX = 1 : 2\), point \(P\) divides \(OX\) in ratio 1:2 from \(O\), so \[ \overrightarrow{OP} = \frac{1}{1+2} a = \frac{1}{3} a \] - Since \(OR : RY = 1 : 3\), point \(R\) divides \(OY\) in ratio 1:3 from \(O\), so \[ \overrightarrow{OR} = \frac{1}{1+3} b = \frac{1}{4} b \] 4. **Find vectors \(\overrightarrow{ZP}\) and \(\overrightarrow{ZR}\):** - \(\overrightarrow{ZP} = \overrightarrow{OP} - \overrightarrow{OZ} = \frac{1}{3} a - (a + b) = \frac{1}{3} a - a - b = -\frac{2}{3} a - b \) - \(\overrightarrow{ZR} = \overrightarrow{OR} - \overrightarrow{OZ} = \frac{1}{4} b - (a + b) = \frac{1}{4} b - a - b = -a - \frac{3}{4} b \) 5. **Find the magnitudes of \(\overrightarrow{ZP}\) and \(\overrightarrow{ZR}\) to determine the ratio \(ZP : ZR\):** Since \(a\) and \(b\) are vectors, the ratio depends on their magnitudes and directions. However, the problem likely expects the ratio of lengths along the line segment \(PR\) or a scalar ratio. 6. **Alternative approach using section formula:** Since \(Z = X + Y - O = a + b\), and points \(P\) and \(R\) are on \(OX\) and \(OY\), the vectors \(ZP\) and \(ZR\) can be expressed as: \[ ZP = |\overrightarrow{Z} - \overrightarrow{P}| = |(a + b) - \frac{1}{3} a| = |\frac{2}{3} a + b| \] \[ ZR = |\overrightarrow{Z} - \overrightarrow{R}| = |(a + b) - \frac{1}{4} b| = |a + \frac{3}{4} b| \] 7. **Express ratio \(ZP : ZR\):** \[ ZP : ZR = |\frac{2}{3} a + b| : |a + \frac{3}{4} b| \] 8. **Simplify ratio if \(a\) and \(b\) are perpendicular and of equal length (common assumption):** Assuming \(|a| = |b| = 1\) and \(a \perp b\), then \[ |\frac{2}{3} a + b| = \sqrt{\left(\frac{2}{3}\right)^2 + 1^2} = \sqrt{\frac{4}{9} + 1} = \sqrt{\frac{13}{9}} = \frac{\sqrt{13}}{3} \] \[ |a + \frac{3}{4} b| = \sqrt{1^2 + \left(\frac{3}{4}\right)^2} = \sqrt{1 + \frac{9}{16}} = \sqrt{\frac{25}{16}} = \frac{5}{4} \] 9. **Calculate ratio:** \[ ZP : ZR = \frac{\sqrt{13}}{3} : \frac{5}{4} = \frac{\sqrt{13}}{3} \times \frac{4}{5} = \frac{4 \sqrt{13}}{15} \] 10. **Final answer:** \[ ZP : ZR = \frac{4 \sqrt{13}}{15} \] This is the simplest form of the ratio given the assumptions. If \(a\) and \(b\) are not perpendicular or equal length, the ratio depends on their magnitudes and angle.