Two geological field teams are working in a remote area. A global positioning system (GPS) tracker at their base camp shows the location of the first team as 37 km away, 24° north of west, and the second team as 34 km away, 38° east of north. When the first team uses its GPS to check the position of the second team, what does it give for the second team's (a) distance from them and (b) direction, measured from due east?
OA = [37cos(24°), 37sin(24°)] km
OB = [-34sin(38°), 34cos(38°)] km
AB = [-34sin(38°)-37cos(24°), 34cos(38°)-37sin(24°)] km
Determine the magnitude and angle (from the east) of this vector.
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To find the distance and direction of the second team from the first team, we can use vector addition and trigonometry.
(a) Distance from the first team:
From the information given, the first team is 37 km away, 24° north of west, and the second team is 34 km away, 38° east of north.
To find the distance between the two teams, we can use the Pythagorean theorem. The horizontal distance (west-east) can be found by adding the horizontal components of the two vectors, and the vertical distance (north-south) can be found by adding the vertical components.
Horizontal distance:
The horizontal component of the first team's vector is 37 km * sin(24°) = 15.628 km west.
The horizontal component of the second team's vector is 34 km * sin(38°) = 20.726 km north.
To find the resultant horizontal distance, we can subtract the westward component from the eastward component:
Horizontal distance = 20.726 km - 15.628 km = 5.098 km north.
Vertical distance:
The vertical component of the first team's vector is 37 km * cos(24°) = 33.156 km north.
The vertical component of the second team's vector is 34 km * cos(38°) = 27.108 km north.
To find the resultant vertical distance, we can add the northward components:
Vertical distance = 33.156 km + 27.108 km = 60.264 km north.
Now, we can find the distance between the two teams using the Pythagorean theorem:
Distance = √((horizontal distance)^2 + (vertical distance)^2)
Distance = √((5.098 km)^2 + (60.264 km)^2)
Distance ≈ 61.4 km
Therefore, the distance of the second team from the first team is approximately 61.4 km.
(b) Direction from due east:
To find the direction, measured from due east, we can use trigonometry. We will use the arctan function to find the angle.
tan(angle) = (opposite / adjacent)
tan(angle) = (horizontal distance / vertical distance)
Using the values we calculated earlier:
tan(angle) = (5.098 km / 60.264 km)
angle ≈ 4.859°
Therefore, the direction of the second team from the first team, measured from due east, is approximately 4.859°.