Tool to calculate the norm of a vector. The vector standard of a vector space represents the length (or distance) of the vector.

Vector Norm - dCode

Tag(s) : Matrix

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Tool to calculate the norm of a vector. The vector standard of a vector space represents the length (or distance) of the vector.

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

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

__Example:__ $ \vec(v) = \left( \begin{array}{c} 1 \ 2 \end{array} \right) $ so $ \|\vec(v)\| = \sqrt{1^2+2^2} = \sqrt{5} $

In space, for a vector $ \vec(v) = (x,y,z) $ the formula is simplified $ \|{\vec {v}}\|= \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} $

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