Definition:Vector Length
Definition
The length of a vector $\mathbf v$ in a normed vector space $\struct {V, \norm {\, \cdot \,} }$ is defined as $\norm {\mathbf v}$, the norm of $\mathbf v$.
Arrow Representation
Let $\mathbf v$ be a vector quantity represented as an arrow in a real vector space $\R^n$.
The length of $\mathbf v$ is the length of the line segment representing $\mathbf v$ in $\R^n$.
Examples
Real Number Line
Let $\mathbf v$ be a vector represented by an arrow on the real number line.
Let:
- the initial point of $\mathbf v$ be $a \in \R$
- the terminal point of $\mathbf v$ be $b \in \R$
The length of $\mathbf v$ is defined as:
- $\norm {\mathbf v} = \size {b - a}$
the absolute value of $b - a$.
Real Vector Space
Let $\mathbf v$ be a vector represented in the real $n$-space $\R^n$ by an ordered $n$-tuple of components $\tuple {v_1, v_2, \ldots, v_n}$.
The length of $\mathbf v$ is defined as:
- $\norm {\mathbf v} = \ds \sqrt {\sum_{i \mathop = 1}^n v_i^2}$
Complex Plane
Let $\mathbf v$ be a vector represented in the complex plane $\C$ by the complex number $z = a + b i$.
The length of $\mathbf v$ is defined as:
- $\norm {\mathbf v} = \cmod z$
where $\cmod z = \sqrt {a^2 + b^2}$ is the modulus of $z$.
Also denoted as
$\size {\mathbf v}$ is often also seen for the length of $\mathbf v$.
Some authorities state that this is not recommended since it can lead to confusion with absolute value.
$\mathsf{Pr} \infty \mathsf{fWiki}$ is less convinced that this is a problem, and both notations can be found here.
It is commonplace to use the non-bold italic form of the symbol for the length, hence using $v$ for the length of $\mathbf v$.
Some sources use $\bmod \mathbf v$ for the length of $\mathbf v$.
Also known as
The length of a vector is also referred to as its module in some older books.
Also see
Sources
- 1921: C.E. Weatherburn: Elementary Vector Analysis ... (previous) ... (next): Chapter $\text I$. Addition and Subtraction of Vectors. Centroids: Definitions: $3$. Definitions of terms
- 1992: Frederick W. Byron, Jr. and Robert W. Fuller: Mathematics of Classical and Quantum Physics ... (previous) ... (next): Volume One: Chapter $1$ Vectors in Classical Physics: $1.2$ The Resolution of a Vector into Components