805 8.3 Geometrically Defined Vectors and Applications Navigation Applications Problems that involve bearing can also be solved using vectors. EXAMPLE 5 Applying Vectors to a Navigation Problem A ship leaves port on a bearing of 28.0° and travels 8.20 mi. The ship then turns due east and travels 4.30 mi. How far is the ship from port? What is its bearing from port? SOLUTION In Figure 30, vectors PA and AE represent the ship’s path. The magnitude and bearing of the resultant PE can be found as follows. Triangle PNA is a right triangle, so angle NAP = 90° - 28.0° = 62.0°, and angle PAE = 180° - 62.0° = 118.0°. Use the law of cosines to find PE , the magnitude of vector PE. PE 2 = 8.202 + 4.302 - 218.20214.302 cos 118.0° Law of cosines PE 2 ≈118.84 Use a calculator. PE ≈10.9 Square root property The ship is about 10.9 mi from port. To find the bearing of the ship from port, find angle APE. sin APE 4.30 = sin 118.0° 10.9 Law of sines sin APE = 4.30 sin 118.0° 10.9 Multiply by 4.30. APE ≈20.4° Use the inverse sine function. Finally, 28.0° + 20.4° = 48.4°, so the bearing is 48.4°. S Now Try Exercise 45. N A P E 4.30 8.20 28.0° 118.08 Figure 30 N Course and ground speed (actual direction of plane) Wind direction and speed Bearing and airspeed Drift angle Figure 31 In air navigation, the airspeed of a plane is its speed relative to the air, and the ground speed is its speed relative to the ground. Because of wind, these two speeds are usually different. The ground speed of the plane is represented by the vector sum of the airspeed and windspeed vectors. See Figure 31. EXAMPLE 6 Applying Vectors to a Navigation Problem An airplane that is following a bearing of 239° at an airspeed of 425 mph encounters a wind blowing at 36.0 mph from a direction of 115°. Find the resulting bearing and ground speed of the plane. SOLUTION An accurate sketch is essential to the solution of this problem. We have included two sets of geographical axes, which enable us to determine measures of necessary angles. Analyze Figure 32 on the next page carefully.
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