Two bicyclists, starting at the same place, are riding toward the same campground by two different routes. One cyclist rides 1500 m due east and then turns due north and travels another 1430 m before reaching the campground. The second cyclist starts out by heading due north for 1870 m and then turns and heads directly toward the campground.

To find the distance between the two cyclists, we can break down their routes into vectors.

Let's first consider the route of the first cyclist. The cyclist first rides 1500 m due east, which can be represented by a vector of magnitude 1500 m and direction 0° (east). Then the cyclist turns due north and rides another 1430 m, which can be represented by a vector of magnitude 1430 m and direction 90° (north). This gives us two vectors:

1. Vector A: 1500 m at 0° (east)
2. Vector B: 1430 m at 90° (north)

For the second cyclist, they start by heading due north for 1870 m, which can be represented by a vector of magnitude 1870 m and direction 90° (north). Then they head directly toward the campground, which means they are heading due east. Let's represent these two vectors:

3. Vector C: 1870 m at 90° (north)
4. Vector D: unknown magnitude at 0° (east)

Now, we need to find the resultant vectors for each cyclist.

To find the resultant vector for the first cyclist (vector AB), we can use the Pythagorean theorem. The horizontal component (east) and vertical component (north) of the resultant vector can be found by adding the corresponding components of vectors A and B:

Horizontal component = 1500 m
Vertical component = 1430 m

Using the Pythagorean theorem, we can find the magnitude of vector AB:

|AB| = √(1500^2 + 1430^2)

To find the resultant vector for the second cyclist (vector CD), we only need to consider vector D since vector C was already in the north direction. We can use the horizontal component of vector C (1870 m) as the magnitude and direction (0°) for vector D.

|CD| = 1870 m

Now, we have the magnitudes of the resultant vectors for both cyclists. The distance between the two cyclists is the difference between these magnitudes:

Distance = |AB| - |CD|

To find out which cyclist is closer to the campground, we can calculate the total distance each cyclist traveled.

For the first cyclist:
Distance in the east direction = 1500 m
Distance in the north direction = 1430 m

To find the total distance, we can use the Pythagorean theorem, since the distance traveled in each direction forms a right triangle.

Total distance for the first cyclist = √((1500^2) + (1430^2))
Total distance for the first cyclist = √(2250000 + 2044900)
Total distance for the first cyclist ≈ √4294900
Total distance for the first cyclist ≈ 2071.35 m (rounded to two decimal places)

For the second cyclist:
Distance in the north direction = 1870 m

Since the second cyclist heads directly toward the campground after traveling north, there is no need to calculate the distance using the Pythagorean theorem.

Total distance for the second cyclist = 1870 m

Comparing the distances traveled, we can see that the second cyclist is closer to the campground, as they traveled a shorter distance of 1870 m compared to the first cyclist's distance of approximately 2071.35 m.