1. **Problem statement:** Two pilots start from point P on a spherical Earth (radius 6378.1 km). Pilot 1 flies at initial bearing N 34° E to point Q, and Pilot 2 flies at initial bearing S 78° E to point R. Both fly at 790 km/h for 90 minutes. We need to find: (a) The great circle distance from Q to R. (b) The bearing from R to Q. 2. **Calculate distance each pilot travels:** Speed = 790 km/h, Time = 90 minutes = 1.5 hours. Distance traveled by each pilot = speed × time = $790 \times 1.5 = 1185$ km. 3. **Convert distances to central angles on the sphere:** Central angle $\theta = \frac{\text{arc length}}{\text{radius}} = \frac{1185}{6378.1} \approx 0.1859$ radians. 4. **Convert initial bearings to azimuths:** - Pilot 1: N 34° E means azimuth $= 34^\circ$. - Pilot 2: S 78° E means azimuth $= 180^\circ - 78^\circ = 102^\circ$ (measured clockwise from north). 5. **Find coordinates of Q and R on the unit sphere:** Using spherical coordinates where latitude $\phi = 90^\circ - \theta$ and longitude $\lambda$ is azimuth. For point Q: $\phi_Q = 90^\circ - (0.1859 \times \frac{180}{\pi}) = 90^\circ - 10.65^\circ = 79.35^\circ$ $\lambda_Q = 34^\circ$ For point R: $\phi_R = 90^\circ - 10.65^\circ = 79.35^\circ$ $\lambda_R = 102^\circ$ 6. **Convert to Cartesian coordinates (unit sphere):** $$x = \cos \phi \cos \lambda, \quad y = \cos \phi \sin \lambda, \quad z = \sin \phi$$ Convert degrees to radians for calculations: $\phi_Q = 79.35^\circ = 1.385$ rad, $\lambda_Q = 34^\circ = 0.593$ rad $\phi_R = 79.35^\circ = 1.385$ rad, $\lambda_R = 102^\circ = 1.780$ rad Calculate: $Q = (\cos 1.385 \cos 0.593, \cos 1.385 \sin 0.593, \sin 1.385) \approx (0.187, 0.115, 0.981)$ $R = (\cos 1.385 \cos 1.780, \cos 1.385 \sin 1.780, \sin 1.385) \approx (-0.167, 0.254, 0.981)$ 7. **Calculate the central angle between Q and R:** Dot product $Q \cdot R = 0.187 \times (-0.167) + 0.115 \times 0.254 + 0.981 \times 0.981 = -0.031 + 0.029 + 0.962 = 0.960$ Central angle $\alpha = \arccos(0.960) = 0.283$ radians 8. **Calculate great circle distance Q to R:** Distance $= \alpha \times$ Earth radius $= 0.283 \times 6378.1 = 1805$ km (rounded to nearest whole number) 9. **Find bearing from R to Q:** Using spherical trigonometry, bearing $\beta$ from R to Q is given by: $$\tan \beta = \frac{\sin(\lambda_Q - \lambda_R) \cos \phi_Q}{\cos \phi_R \sin \phi_Q - \sin \phi_R \cos \phi_Q \cos(\lambda_Q - \lambda_R)}$$ Calculate numerator: $\sin(0.593 - 1.780) \cos 1.385 = \sin(-1.187) \times 0.187 = -0.927 \times 0.187 = -0.173$ Calculate denominator: $\cos 1.385 \sin 1.385 - \sin 1.385 \cos 1.385 \cos(0.593 - 1.780) = 0.187 \times 0.981 - 0.981 \times 0.187 \times \cos(-1.187)$ $= 0.183 - 0.981 \times 0.187 \times 0.375 = 0.183 - 0.069 = 0.114$ Therefore: $\tan \beta = \frac{-0.173}{0.114} = -1.52$ $\beta = \arctan(-1.52) = -56.7^\circ$ Since bearing is clockwise from north, add 360° if negative: $\beta = 360 - 56.7 = 303.3^\circ$ **Final answers:** (a) Distance from Q to R along great circle $\approx 1805$ km. (b) Bearing from R to Q $\approx 303.3^\circ$.