1. **State the problem:** You want to fly North relative to a stationary observer while there is a wind blowing from the West at 15 km/h. Your airspeed (speed relative to the air) is 90 km/h. We need to find the bearing (degrees clockwise from North) you must fly to achieve this. 2. **Define variables:** - Let $\vec{v}_a$ be your velocity relative to the air (your flying direction and speed). - Let $\vec{v}_w$ be the wind velocity (15 km/h from the West means wind vector points East). - Let $\vec{v}_g$ be your ground velocity (velocity relative to the stationary observer), which we want to be due North. 3. **Set up vectors:** - Wind velocity: $\vec{v}_w = (15, 0)$ km/h (East is positive x-direction). - Ground velocity desired: $\vec{v}_g = (0, v_g)$ km/h (North is positive y-direction). - Air velocity: $\vec{v}_a = (v_{ax}, v_{ay})$ with magnitude $|\vec{v}_a| = 90$ km/h. 4. **Relation between velocities:** $$\vec{v}_g = \vec{v}_a + \vec{v}_w$$ Since $\vec{v}_g = (0, v_g)$ and $\vec{v}_w = (15, 0)$, we have: $$ (0, v_g) = (v_{ax}, v_{ay}) + (15, 0) $$ This implies: $$ v_{ax} + 15 = 0 \Rightarrow v_{ax} = -15 $$ $$ v_{ay} = v_g $$ 5. **Use magnitude of air velocity:** $$ |\vec{v}_a| = \sqrt{v_{ax}^2 + v_{ay}^2} = 90 $$ Substitute $v_{ax} = -15$: $$ \sqrt{(-15)^2 + v_{ay}^2} = 90 $$ $$ \sqrt{225 + v_{ay}^2} = 90 $$ Square both sides: $$ 225 + v_{ay}^2 = 8100 $$ $$ v_{ay}^2 = 8100 - 225 = 7875 $$ $$ v_{ay} = \sqrt{7875} \approx 88.7 \text{ km/h} $$ 6. **Calculate bearing:** The bearing is the angle clockwise from North to your flying direction. Your air velocity components are: - $v_{ax} = -15$ (Westward) - $v_{ay} = 88.7$ (Northward) The angle $\theta$ from North towards West is: $$ \theta = \arctan\left(\frac{|v_{ax}|}{v_{ay}}\right) = \arctan\left(\frac{15}{88.7}\right) $$ Calculate: $$ \theta \approx \arctan(0.169) \approx 9.59^\circ $$ Since $v_{ax}$ is negative (West), the direction is $9.59^\circ$ West of North. 7. **Convert to bearing (degrees clockwise from North):** Bearing = $360^\circ - 9.59^\circ = 350.4^\circ$ (to 3 significant figures) **Final answer:** You must fly on a bearing of approximately **350 degrees** to travel due North relative to the ground in the presence of the wind.