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Vector Norm

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

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Vector Norm -

Tag(s) : Matrix

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# Vector Norm

## Vector's Norm Calculator

### What is the definition of a norm of a 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).

### How to calculate the norm of a vector?

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

### How to calculate the components of a vector from the points?

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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