villages A;B;C;D are such that B is 4km due east of A ;C is 3km due south of B and D is 4km S50w from C.Calculate the distance and bearing of A from D.
with A at (0,0) we just add the x- and y-displacements to get the final location:
<4,0>+<0,-3>+<-3.06,-2.57> = <0.94,-5.57>
Now you can find the distance and bearing.
To calculate the distance and bearing of A from D, we can use vector addition and trigonometry.
First, let's break down the given information into vectors:
1. The vector from A to B is 4 km due east.
2. The vector from B to C is 3 km due south.
3. The vector from C to D is 4 km S50W (south 50 degrees west).
Now, let's calculate the position vector from A to D by adding these vectors.
Since B is east of A and C is south of B, the vector from A to C can be obtained by adding the vectors from A to B and from B to C.
1. The vector from A to C = (4 km due east) + (3 km due south)
To add these vectors, we need to break them down into their x and y components:
- The vector from A to C, in terms of its x and y components, is (4 km, -3 km).
Next, we need to find the vector from C to D, which is 4 km S50W.
To convert this vector into its x and y components, we can use trigonometry. The angle S50W can be broken down into its north and west components.
1. The north component = 4 km * sin(50 degrees)
2. The west component = 4 km * cos(50 degrees)
These north and west components give us the vector from C to D in terms of its x and y components:
- The vector from C to D = (-2.576 km, -2.828 km)
Now, we can find the vector from A to D by adding the vectors from A to C and from C to D:
1. The vector from A to D = (4 km, -3 km) + (-2.576 km, -2.828 km)
By adding the respective components, we get:
- The vector from A to D = (1.424 km, -5.828 km)
Finally, we can use the Pythagorean theorem to find the distance between A and D:
Distance = √((1.424 km)^2 + (-5.828 km)^2)
Distance ≈ 6.020 km
To calculate the bearing of A from D, we can use trigonometry again.
The bearing is the angle between the north direction and the line connecting A and D. We can calculate this angle by taking the arctan of the ratio of the y-component to the x-component of the vector from A to D:
Bearing = arctan((-5.828 km) / (1.424 km))
Bearing ≈ -75.06 degrees
Therefore, the distance from A to D is approximately 6.020 km, and the bearing of A from D is approximately -75.06 degrees.