1. **Problem statement:** We need to find the displacement vector from Winnipeg to Saskatoon given two vectors: Winnipeg to Regina (530 km at 6.9° north of west) and Regina to Saskatoon (230 km at 53° north of west). 2. **Understanding the directions:** "North of west" means we start facing west (which is 180° from east) and rotate northwards by the given angle. So the angles from the positive x-axis (east) are: - Winnipeg to Regina: $180^\circ - 6.9^\circ = 173.1^\circ$ - Regina to Saskatoon: $180^\circ - 53^\circ = 127^\circ$ 3. **Vector components:** We convert each vector into components using: $$x = r \cos(\theta), \quad y = r \sin(\theta)$$ where $r$ is the magnitude and $\theta$ is the angle from the positive x-axis. 4. **Calculate components for Winnipeg to Regina:** $$x_1 = 530 \cos(173.1^\circ) = 530 \times \cos(173.1^\circ)$$ $$y_1 = 530 \sin(173.1^\circ) = 530 \times \sin(173.1^\circ)$$ 5. **Calculate components for Regina to Saskatoon:** $$x_2 = 230 \cos(127^\circ) = 230 \times \cos(127^\circ)$$ $$y_2 = 230 \sin(127^\circ) = 230 \times \sin(127^\circ)$$ 6. **Sum components to get Winnipeg to Saskatoon displacement:** $$x = x_1 + x_2$$ $$y = y_1 + y_2$$ 7. **Calculate numerical values:** Using approximate cosine and sine values: - $\cos(173.1^\circ) \approx -0.993$, - $\sin(173.1^\circ) \approx 0.120$, - $\cos(127^\circ) \approx -0.601$, - $\sin(127^\circ) \approx 0.799$. Calculate: $$x_1 = 530 \times (-0.993) = -526.29$$ $$y_1 = 530 \times 0.120 = 63.60$$ $$x_2 = 230 \times (-0.601) = -138.23$$ $$y_2 = 230 \times 0.799 = 183.77$$ Sum: $$x = -526.29 + (-138.23) = -664.52$$ $$y = 63.60 + 183.77 = 247.37$$ 8. **Find magnitude of displacement:** $$d = \sqrt{x^2 + y^2} = \sqrt{(-664.52)^2 + (247.37)^2}$$ $$d = \sqrt{441572 + 61192} = \sqrt{502764} \approx 709.06 \text{ km}$$ 9. **Find direction angle north of west:** Angle $\phi$ from west towards north is: $$\phi = \tan^{-1}\left(\frac{y}{-x}\right) = \tan^{-1}\left(\frac{247.37}{664.52}\right)$$ $$\phi \approx \tan^{-1}(0.372) = 20.4^\circ$$ **Final answer:** The displacement of Saskatoon from Winnipeg is approximately 709 km at 20.4° north of west.