two geolocial field temas are working in an area a GPS at their base camp shows a location of the first team as 42 km away 22 degrees north of west, and the second team as 30 km away 31 degrees east of north when the first team uses the GPS to check the position of the second team what does it give for the second teams distance from them and direction measure due east. draw a graph as well that is 2d measuring the numbers that is very simple

Let's break this down step by step and solve it using vector components.

### Step 1: Understanding the given data
- **First team**: 42 km away, 22 degrees north of west.
- **Second team**: 30 km away, 31 degrees east of north.

We are asked to determine the distance and direction of the second team from the first team. To solve this, we’ll place the base camp as the origin \((0, 0)\), and express the positions of both teams in terms of their \(x\)- and \(y\)-coordinates.

### Step 2: Convert both vectors into components
We will break both vectors into their east-west (\(x\)-axis) and north-south (\(y\)-axis) components.

1. **Position of the first team**:
   - Distance: \(42\) km.
   - Angle: \(22^\circ\) north of west.
   - Components:
     - \(x_1 = -42 \cos(22^\circ)\) (negative because it’s west).
     - \(y_1 = 42 \sin(22^\circ)\) (positive because it’s north).

2. **Position of the second team**:
   - Distance: \(30\) km.
   - Angle: \(31^\circ\) east of north.
   - Components:
     - \(x_2 = 30 \sin(31^\circ)\) (positive because it’s east).
     - \(y_2 = 30 \cos(31^\circ)\) (positive because it’s north).

### Step 3: Calculate the relative position of the second team from the first team
To find the second team’s position relative to the first team, we need the difference in their \(x\)- and \(y\)-coordinates:

\[
\Delta x = x_2 - x_1
\]
\[
\Delta y = y_2 - y_1
\]

### Step 4: Distance and direction
- The distance between the two teams can be found using the Pythagorean theorem:
  \[
  d = \sqrt{(\Delta x)^2 + (\Delta y)^2}
  \]
- The direction, relative to due east, can be found by calculating the angle \(\theta\):
  \[
  \theta = \tan^{-1}\left(\frac{\Delta y}{\Delta x}\right)
  \]

### Step 5: Drawing the graph
I'll plot a simple 2D graph showing the base camp at the origin, and the positions of the first and second teams with arrows representing the vectors.

Let me calculate the components and plot the graph.

Two geological field teams are working in an area. A GPS at their base camp shows a location of the first team as 42 km away, 22 degrees north of west, and the second team as 30 km away, 31 degrees east of north. When the first team uses the GPS to check the position of the second team, what does it give for the second team's distance from them and direction measured due east? Draw a graph as well that is 2D, measuring the numbers in a simple manner.

Solve a problem where two geological field teams are located at different points, and we need to calculate the distance and direction between them using vector components and trigonometry. Includes step-by-step breakdown and formulas. Suitable for students in Grades 10-12.

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