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# Points, Vectors, and Functions

# 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 *x*^{2} + *y*^{2} = 1.

However, if we want to draw a vertical line, *x* = 2 is much nicer than r = 2 / cos θ.

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.