Vector Norm

The norm of a vector is a function that associates each vector with its length, that is, a non-negative real number. This norm is called the Euclidean norm or quadratic norm, or L² norm.

In a space of dimension $ n $, if $ A $ and $ B $ are two points, the norm of the vector $ \overrightarrow{AB} $, denoted $ \|\overrightarrow{AB}\| $, corresponds to the distance between $ A $ and $ B $ (the length of the segment $ [AB] $).

In dimension 1, the norm reduces to the absolute value of the real number.

In a Euclidean vector space of dimension $ n $, the norm of a vector $ \vec{v} = (x_1, x_2, \dots, x_n) $ is calculated by the square root of the sum of the squares of its components:

$$ \| \vec{v} \| = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2} $$

It can also be calculated from the dot product of the vector with itself: $$ \| \vec{v} \| = \sqrt{ \vec{v} \cdot \vec{v} } $$

Normalizing a vector involves constructing a unit vector (of magnitude 1) with the same direction as the original vector.

For a vector $ \vec{v} $, normalization is performed by dividing the vector by its norm: $$ \vec{u} = \frac{\vec{v}}{\|\vec{v}\|} $$

This operation is only possible if the vector is not zero: $ \vec{v} \neq \vec{0} $

In the 2D plane, for a vector $ \vec{v} = (x,y) $ the formula is simplified $$ \|\vec{v}\|= \sqrt{x^2+y^2} $$

In 3D space, for a vector $ \vec{u} = (x,y,z) $ the formula is simplified $$ \|{\vec{u}}\|= \sqrt{x^2+y^2+z^2} $$

Geometrically, this norm corresponds to the distance of the point from the origin $ (0,0,0) $

From the coordinates of the points $ A (x_A,y_A) $ and $ B (x_B,y_B) $ of the vector $ \overrightarrow{AB} $, the components of the vector are $ {\overrightarrow {AB}} = \{ (x_B-x_A), (y_B-y_A) \} $ and therefore the norm is $ \|\overrightarrow {AB}\| = \sqrt{(x_B-x_A)^2+(y_B-y_A)^2} $

This generalizes to dimension $ n $, the components are $$ \overrightarrow{AB} = (x_{B1} - x_{A1}, \; x_{B2} - x_{A2}, \; \dots, \; x_{Bn} - x_{An}) $$ and the norm $$ \|\overrightarrow{AB}\| = \sqrt{(x_{B1} - x_{A1})^2 + (x_{B2} - x_{A2})^2 + \cdots + (x_{Bn} - x_{An})^2} $$

A unit vector is a vector with a norm equal to 1: $ \| \vec{v} \| = 1 $