TITLE: What are vector norms used for?
QUESTION [7 upvotes]: I'm currently working with a computer science problem that requires me to build vectors that can return their own norms. Based on Wolfram Alpha's description, I think I have an idea of how this is accomplished for the simple $L^2$ norm ($\sqrt{a^2+b^2+c^2}$, and so on) required by the exercise, but I've no notion of what this is actually useful for or why I would want to find it, outside of it being required by the exercise. 
Any insight is appreciated!

REPLY [5 votes]: Norms are a measure of distance. One has different ways to define what is the distance between points in multiple dimensions, which collapse to the usual notion of the absolute value in 1D.
in particular, Euclidean distance is defined by the 2-norm,
$$
\left\| \begin{pmatrix} x_1 \\ \vdots \\ x_k \end{pmatrix} \right\|_2
 = \sqrt{\sum_{i=1}^k x_i^2}
$$
There are others, of which the 1-norm
$$
\left\| \begin{pmatrix} x_1 \\ \vdots \\ x_k \end{pmatrix} \right\|_1
 = \sum_{i=1}^k \left|x_i^2\right|
$$
and the infinity norm
$$
\left\| \begin{pmatrix} x_1 \\ \vdots \\ x_k \end{pmatrix} \right\|_\infty
 = \max_{1 \le i \le k} \left\{|x_i|\right\}
$$
are the most useful.
You are welcome to read up more on Wikipedia