At a Glance - In the Real World
You might feel like you're taking a foreign language class, calling the same point by polar coordinates (r, θ) or rectangular coordinates (x, y). Maybe we're talking about a vector, in which case |r| is its magnitude, θ is its direction, and x and y are its components.
The reason for all the choices is that different representations are useful for different tasks. If we want to draw the unit circle,
r = 1 is a much nicer equation than x2 + y2 = 1.
However, if we want to draw a vertical line, x = 2 is much nicer than
Taking things a little closer to the real world, if we're trying to gravel from point A to B in downtown New York we need to use the Manhattan distance to compute the distance. We need to know the x and y components of the vector between points A and B.
However, if we're planning to fly things from point A to point B in the middle of nowhere, we consider the distance as the crow flies. We want to know the straight-line distance from A to B, which is the magnitude of the vector connecting those points.