Length of a vector

Definition

Let $n$ be a positive integer and let $v = (v_1, \dots, v_n)$ be a vector in $\mathbb{R}^n$ . The length of $v$ is the real number $\|v\| = \sqrt{v_1^2 + v_2^2 + \dots + v_n^2} = \sqrt{v \cdot v},$ where $v \cdot v = v_1^2 + \dots + v_n^2$ is the dot product of $v$ with itself. A unit vector is a vector of length 1.

Remarks

For each vector $v$ , $\|v\| \geq 0$ , and $\|v\| = 0$ if and only if $v$ is the zero vector. For each real number $c$ , $\|c v\| = |c| \, \|v\|$ . Thus for $v \neq 0$ the vector $v / \|v\|$ is a unit vector. Other names for the length are the magnitude and the Euclidean norm. The notation $|v|$ is also common.

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Euclidean vector concept
Length of a vector definition

Tag	`0DZS7ZG0QY`
URI	`m2://0DZS7ZG0QY`
Kind	definition
Writing	W0 stub
MSC	51N20
Defines	Length of a vector
Uses	Euclidean vector

Length of a vector

Definition

Remarks

Reading path

Further reading

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