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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

### N-Vector (N-dimensional)

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

### 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 segment).

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

In a 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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