- 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

Vectors aren't always 2-D. We can have an *n*-dimensional vector for any positive integer *n* we like. We could have a 100-dimensional vector, or a googolplex-dimensional vector.

We can also have vectors with infinitely many components.

Look at some examples.

Here's a vector with infinitely many components, all of which are 1: (1,1,1,...)

We could have a vector where the component in the *n*^{th} place is *n*: (1,2,3,...)

We could have a vector whose components alternate between 0 and 1: (0,1,0,1,...)

Is the last component of the vector (0,1,0,1,...) equal to 0 or a 1? Answer. Sorry - that's a trick question. This vector keeps going forever, it doesn't have a last component.

There are quite a few different ways to measure the size of a vector *v* = (*x*_{1}, *x*_{2}, ..., *x*_{n}).

This sort of measurement is called the *taxicab norm* or *Manhattan distance*. For a 2-D vector (*x*,*y*), the Manhattan distance

*x* + *y*

is the distance we need to travel from the tail of the vector to the head of the vector if we need to stay on the grid-like streets.

We've haven't defined the word *norm* yet. A norm is a mathematical thing that has to follow a bunch of fussy rules, but if we think of it as a way to measure size, that's close enough. A mathematician may use one norm or another norm depending on what they're doing with their vectors.

In parting, we saw that it's possible to have infinite-dimensional vectors. How could we measure the size of an infinite-dimensional vector?