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Vector analysis helps us understand situations involving direction and magnitude, like the airplane tracked by a radar station. In this case, the position of the airplane changes from an initial to a final point. This can be described in terms of vectors. A vector has two key properties: its length (magnitude) and where it points (direction).
Vectors can be added, subtracted, and multiplied by scalars. Here, the initial and final positions are vectors. Calculating the displacement means finding the vector that shows the change in position from start to finish.
By converting the airplane's radar coordinates into Cartesian coordinates, we can treat the positions as vectors and use arithmetic operations to get the displacement vector. The displacement is a new vector resulting from subtracting the initial position vector from the final position vector. This gives both the magnitude and direction of the airplane's movement during the tracked period.

The Pythagorean Theorem is crucial in finding the magnitude of the displacement vector. This theorem applies to right-angled triangles and states that the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In our case, after finding the horizontal component (\(\Delta x\)) and the vertical component (\(\Delta h\)) of the airplane's displacement vector, we can think of these as the two sides of a right triangle. The total displacement (\(\Delta d\)) is like the hypotenuse.
Using the Pythagorean theorem, the magnitude of displacement is calculated as follows:

• The formula is \(\Delta d = \sqrt{(\Delta x)^2 + (\Delta h)^2}\).
• This yields a single number representing how far the airplane has moved, regardless of the path it took.

Understanding this theorem is key to solving many problems that involve distances and coordinates, helping us find how far an object traveled in linear space.

Cartesian coordinates are a system that uses two numbers to pinpoint a position on a plane, much like a map. These numbers are usually denoted as \(x\) and \(y\), representing horizontal and vertical positions respectively.
For the radar problem, the airplane's position is first described using polar coordinates: distance from the radar and angle above the horizon. We convert these to Cartesian coordinates to simplify the vector calculation process.

• To find the initial position: - The vertical position (height \(h_1\)) is calculated with \(h_1 = d_1 \sin(\theta_1)\). - The horizontal position (\(x_1\)) is obtained by \(x_1 = d_1 \cos(\theta_1)\).
• Similarly, the final position is calculated using the same method with different distance and angle values.

Understanding how to switch between polar and Cartesian coordinates helps in tackling various problems in physics, engineering, and mathematics, allowing for easier calculations and visualizations of movement and change.