The Vector Norm
In mathematics, a norm is a function that assigns a length to a vector.
Definition of a Norm
In a real vector space V over the field K=R, a norm on V is a function ||·||:V→R such that: $$||v|| \ge 0 \;\;\; \forall v ∈ V \\ ||v||=0 \:\:\:\:\: if \: and \: only \: if \:\: V=0_v \\ ||k \cdot v||=|k|·||v|| \:\:\: \forall v \in V , k \in R \\ ||v1+v2|| ≤ ||v_1||+||v_2|| \:\:\: \forall v_1,v_2 \in V $$
The norm ||v|| of a vector v is referred to as the magnitude or length of v.
Types of Norms
There are various types of norms:
$$ ||v||_1 := \sum_i^n |x_i| \:\:\:\: \forall v = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}$$
$$ ||v||_2 := \sqrt{\sum_i^n x^2_i} \:\:\:\: \forall v = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}$$
Note. This latter is known as the Euclidean norm. It's the most commonly used since it describes operations in Euclidean geometry.
$$ ||v||_{\infty} := max \{ |x_i| \} \:\:\:\: \forall v = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}$$
Depending on the chosen norm, the length of the vector varies.
A Practical Example
Consider a vector v in the vector space V=R3
$$ v = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}= \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} $$
The magnitude (length) of the vector takes the following values depending on the chosen norm:
$$ ||v||_1 := |x_1|+|x_2|+|x_3| = |2|+|3|+|4| = 9 $$
$$ ||v||_2 := \sqrt{x_1^2+x_2^2+x_3^2} = \sqrt{2^2+3^2+4^2} = \sqrt{29} $$
$$ ||v||_{\infty} := max \{ x_1, x_2, x_3 \} = max \{ 2, 3, 4 \} = 4 $$
The Unit Vector or Direction Vector
A unit vector (or direction vector) is a vector with a magnitude equal to 1 $$ ||v|| = 1 $$
Each vector vi in the vector space is associated with a direction vector ||vi||
Example
In the Euclidean norm, the vector v (2,3,4) is associated with the direction vector ||v||=√29
$$ v = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} $$
$$ ||v|| = \sqrt{29} $$
To obtain the unit (direction) vector of v, simply divide the vector by its norm.
$$ \hat{v} = \frac{1}{||v||} $$
$$ \hat{v} = \frac{1}{ \sqrt{29} } \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} $$
$$ \hat{v} = \begin{pmatrix} \frac{2}{ \sqrt{29} } \\ \frac{3}{ \sqrt{29} } \\ \frac{4}{ \sqrt{29} } \end{pmatrix} $$
This results in the direction vector of vector v.
Note. The transition from vector to direction vector is called the normalization of the vector.
