An aircraft flies around a triangular course. The first leg is 200 km on a bearing of 1150 and the second leg is150 km on a bearing of 2300
. How long is the third leg of the course and on what bearing must the aircraft
fly?
you fly on a heading; you take a bearing.
If we label the triangle ABC and draw a diagram, it is clear that
AC^2 = 200^2 + 150^2 - 2*200*150 cos65°
200cis(-25°) + 150cis(-140°) = -66.35 - 180.9i = 192.68cis(-70°)
so the bearing of A from C is 340°
To find the length of the third leg of the triangular course and the bearing the aircraft must fly, we can use the concept of vector addition.
1. Start by drawing a sketch of the triangle. Label the three vertices A, B, and C, and the lengths of the first and second legs as A-B = 200 km and B-C = 150 km, respectively.
2. Convert the bearings to angles in degrees to work with them easily. The bearing of 1150 is equivalent to 115 degrees, and the bearing of 2300 is equivalent to 230 degrees.
3. Define vectors AB and BC based on the given lengths and angles. To define a vector, we break it down into its horizontal and vertical components. The horizontal component is the length multiplied by the cosine of the angle, and the vertical component is the length multiplied by the sine of the angle.
The vector AB can be defined as AB = (200 * cos(115°), 200 * sin(115°)), and BC can be defined as BC = (150 * cos(230°), 150 * sin(230°)).
4. Add the two vectors AB and BC to find the resultant vector AC. To add two vectors, you add their corresponding components.
AC = AB + BC = (ABx + BCx, ABy + BCy).
Calculate AC by adding the x-components and y-components separately.
AC = (200 * cos(115°) + 150 * cos(230°), 200 * sin(115°) + 150 * sin(230°)).
5. Calculate the magnitude (length) of the vector AC using the Pythagorean theorem.
The magnitude of AC = sqrt((ACx)^2 + (ACy)^2).
6. The magnitude of AC is the length of the third leg of the triangular course. Round the value to the desired precision.
7. To find the bearing of the aircraft, use the inverse tangent function (atan2) to calculate the angle between the x-axis and the vector AC.
The bearing = atan2(ACy, ACx) in degrees.
8. Convert the bearing from a mathematical angle (where 0° is the positive x-axis and angles increase counterclockwise) to the conventional compass bearing.
To convert, subtract the bearing from 90° and add 360° if the result is negative.
That's how you can find the length of the third leg of the course and the bearing the aircraft must fly using vector addition.