Determining Displacement in a Cartesian Plane: A Classic Geometry Problem

Imagine a man embarking on a journey across a vast flat landscape, moving precisely one mile due east, five miles due south, two miles due east, and finally nine miles due north. This journey brings us to a classic problem in geometry: determining displacement. In this article, we will explore how to calculate the distance from the starting point to the final position using the Pythagorean theorem.

The Journey

The man's journey can be broken down into movements that can be plotted on a Cartesian coordinate system:

Starting Point: (0, 0) First Move: 1 mile east → (1, 0) Second Move: 5 miles south → (1, -5) Third Move: 2 miles east → (3, -5) Fourth Move: 9 miles north → (3, 4)

Thus, the final position of the man is at coordinates (3, 4).

Calculating the Distance Using the Pythagorean Theorem

To find the distance from the starting point (0, 0) to the final position (3, 4), we can use the distance formula:

Distance (sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2})

Substituting the coordinates into the formula, we get:

Distance (sqrt{(3 - 0)^2 (4 - 0)^2}) (sqrt{3^2 4^2}) (sqrt{9 16}) (sqrt{25}) 5

Therefore, the man is 5 miles away from the starting point. This straightforward solution applies to a flat, Cartesian plane and assumes that the Earth is a perfect sphere (which it is not).

Considerations for Nearby Poles

However, if the journey takes place near the North or South Pole, the geometry of the Earth comes into play. Near the poles, the curvature of the Earth means that the solution changes:

Case 1: If the man is 8 km from the North Pole when he starts, he can travel 8 km east and then stay exactly 8 km from the North Pole, even as he travels south and north, based on the Earth's curvature.

Case 2: If the man is closer than 8 km to the North Pole, the calculation becomes more complex and potentially unsolvable due to the Earth's spherical geometry, which means the triangle formed by the movements would not be a right triangle in the traditional Euclidean sense.

Considering Routes

The problem's solution also hinges on the movement route taken. For example:

If the man follows the same path back, the total distance would be 14 km (1 5 2 9). If the path is described as A to B to C, then from C to A may result in a different distance. A straight-line path from C to A (the hypotenuse) would be simpler, forming a right-angled triangle.

In this right-angled triangle, the hypotenuse can be calculated as follows:

The square of the hypotenuse is the sum of the squares of the opposite sides:

(6^2 8^2 36 64 100)

The square root of 100 is 10:

Therefore, the man is 10 kilometers from where he started, and he is north-east of where he began.

Conclusion

Understanding the problem of displacement in a Cartesian plane using the Pythagorean theorem provides a straightforward solution. However, when considering the real-world scenario, the solution must account for the Earth's curvature and potential complexities of the path taken. This problem offers a fascinating glimpse into how geometry applies in real-world navigation and problem-solving.