An air-traffic controller observes two aircraft on his radar screen. The first is at altitude 800 m, horizontal distance 19.8 km, and 24.0° south of west. The second aircraft is at altitude 1000 m, horizontal distance 17.0 km, and 12.0° west of south.
What is the displacement vector FROM the first plane TO the second plane, letting represent east, north, and up.
To solve this problem, we need to break down the given information into vector components and then calculate the displacement vector from the first plane to the second plane.
First, let's convert the angles into standard Cartesian coordinate system angles (where east is the positive x-axis, north is the positive y-axis, and up is the positive z-axis).
For the first plane:
- Altitude (z-component): 800 m (positive)
- Horizontal distance (x-component): 19.8 km * cos(24.0°) (positive)
- Horizontal distance (y-component): -19.8 km * sin(24.0°) (negative because south of west)
For the second plane:
- Altitude (z-component): 1000 m (positive)
- Horizontal distance (x-component): -17.0 km * sin(12.0°) (negative because west of south)
- Horizontal distance (y-component): -17.0 km * cos(12.0°) (negative because west of south)
Now, we can calculate the displacement vector from the first plane to the second plane by subtracting the corresponding components:
Displacement in the x-axis: x2 - x1
Displacement in the y-axis: y2 - y1
Displacement in the z-axis: z2 - z1
Substituting the values:
x-displacement: [-17.0 km * sin(12.0°)] - [19.8 km * cos(24.0°)]
y-displacement: [-17.0 km * cos(12.0°)] - [-19.8 km * sin(24.0°)]
z-displacement: 1000 m - 800 m
Finally, the displacement vector from the first plane to the second plane is:
[-17.0 km * sin(12.0°) - 19.8 km * cos(24.0°), -17.0 km * cos(12.0°) + 19.8 km * sin(24.0°), 200 m]
Note: The positive/negative signs and the specific angles used in the calculations may vary depending on the coordinate system convention used.
To find the displacement vector from the first plane to the second plane, we need to calculate the vector between their positions.
First, let's break down the given information:
First Plane:
Altitude: 800 m
Horizontal Distance: 19.8 km
Direction: 24.0° south of west
Second Plane:
Altitude: 1000 m
Horizontal Distance: 17.0 km
Direction: 12.0° west of south
To calculate the displacement vector, we need to find the horizontal and vertical components separately.
For the horizontal component, we can use the cosine rule:
Horizontal Component:
First Plane: 19.8 km * cos(24.0°)
Second Plane: 17.0 km * sin(12.0°)
Vertical Component:
First Plane: 800 m
Second Plane: 1000 m
Now, let's calculate the components:
Horizontal Component:
First Plane: 19.8 km * cos(24.0°) = 17.933 km
Second Plane: 17.0 km * sin(12.0°) = 3.509 km
Vertical Component:
First Plane: 800 m
Second Plane: 1000 m
Now, we have the horizontal and vertical components for the displacement vector.
Displacement Vector (East, North, Up):
Horizontal Component: -17.933 km (negative because it's from the first plane to the second plane)
Vertical Component: 200 m (subtracting the first plane's altitude from the second plane's altitude)
So, the displacement vector from the first plane to the second plane is approximately -17.933 km (East), 3.509 km (North), and 200 m (Up).