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 43 km away, 23° north of west, and the second team as 29 km away, 33° 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?
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To find the distance and direction of the second team from the perspective of the first team, we can use vector addition.
First, let's convert the given information into Cartesian coordinates.
The location of the first team is 43 km away and 23° north of west. To convert this to Cartesian coordinates, we can use trigonometric functions:
x1 = -43 km * sin(23°)
y1 = 43 km * cos(23°)
The location of the second team is 29 km away and 33° east of north. Again, using trigonometric functions:
x2 = 29 km * sin(33°)
y2 = 29 km * cos(33°)
Now, we can calculate the distance and direction from the perspective of the first team:
(a) Distance:
The distance between the two teams can be found using the distance formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
(b) Direction:
The direction can be determined by finding the angle between the positive x-axis and the line connecting the first and second teams. This can be calculated using trigonometry:
direction = atan2(y2 - y1, x2 - x1)
Now, substituting the values we calculated into these formulas, we can find the answers to both (a) and (b).