- Topics At a Glance
- Functions
- Increasing or Decreasing or...
- Bounded
- Even and Odd Functions
- Vectors: A New Kind of Animal
- Magnitude
- Direction
- Scaling Vectors
- Unit Vectors
- Vector Notation
- (More than 2)-Dimensional Vectors
- Vector Functions
- Sketching Vector Functions
- Parametric Equations
- Graphing Parametric Equations
- Points on Graphs of Parametric Equations
- Parametrizations of the Unit Circle
- Parameterization of Lines
- Polar Coordinates
- Simple Polar Inequalities
- Switching Coordinates
- Translating Equations and Inequalities between Coordinate Systems
- Polar Functions
- Graphing Polar Functions
- Rules of Graphing We Do (or Don't) Have
- Bounds on Theta
- Intersections of Polar Functions
**In the Real World**- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

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.