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) : Mathematics, 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 segment).

In a space of dimension n, a vector \( \vec(v) \) of components \( x_i \) : \( \vec(v) = (x_1, x_2, ..., x_n) \) computes 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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