Vector Norm

The norm of a vector is its length. If $ A $ and $ B $ are two points (of a space of $ n $ dimensions) then the norm of the vector, noted with a double bar $ \|\overrightarrow{AB}\| $, is the distance between $ A $ and $ B $ (the length of the segment $ [AB] $).

The absolute value is the special case of the norm for a real number (one dimension).

In a vector space of dimension $ n $, a vector $ \vec{v} $ of components $ x_i $ : $ \vec{v} = (x_1, x_2, ..., x_n) $ is computed by the square root of the sum of the squares of the components: $$ \left\|\vec{v}\right\| = \sqrt{x_1^2 + x_2^2 + \cdots +x_n^2} $$

The norm of a vector can also be computed from the scalar product of the vector with itself: $ \| \vec{v} \| = \sqrt{ \vec{v} \cdot \vec{v} } $.

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 $$ \|{\vec{u}}\|= \sqrt{x^2+y^2+z^2} $$

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} $