Euclidean Norm¶

Definition¶

For a vector $\boldsymbol{x}=(x_1,x_2,\ldots,x_n)\in\mathbb{R}^n$ $R^{n}$ , the Euclidean norm (also called the length or 2-norm) is

\|\boldsymbol{x}\|=\sqrt{\boldsymbol{x}\cdot \boldsymbol{x}}=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}.

A unit vector is a vector with norm $1$ . For nonzero $\boldsymbol{x}$ , the vector $\frac{1}{\|\boldsymbol{x}\|}\boldsymbol{x}$ is the unit vector in the direction of $\boldsymbol{x}$ ; this process is called normalization.

Interpretation¶

The Euclidean norm measures the length of a vector, the distance from the origin to its tip. It extends the absolute value of a number: for $n=1$ $1$ , $\|x\|=\sqrt{x^2}=|x|$ $∣ x ∣$ . Normalization keeps the direction of a nonzero vector but changes its length to $1$ .

A vector v with components v1 and v2 forming a right triangle; its length, the norm, is the hypotenuse.

In the figure the components $v_1$ and $v_2$ are the legs of a right triangle and $\boldsymbol{v}$ is its hypotenuse, so Pythagoras gives $\|\boldsymbol{v}\|=\sqrt{v_1^2+v_2^2}$ $v_{1}^{2} + v_{2}^{2}$ ; for instance $\|(3,4)\|=\sqrt{9+16}=5$ $5$ . In $\mathbb{R}^3$ Pythagoras applied twice gives $\sqrt{v_1^2+v_2^2+v_3^2}$ , and the definition continues the pattern to $n$ coordinates.

Drag $\boldsymbol{x}$ : its norm is the hypotenuse of the right triangle with legs $|x_1|$ and $|x_2|$ . Dividing by $\|\boldsymbol{x}\|$ moves the vector onto the unit circle.

The squares make every term nonnegative, so $\|\boldsymbol{x}\|\ge0$ $0$ , with equality only for $\boldsymbol{x}=\boldsymbol{0}$ $0$ ; the square root brings the result back to the scale of the coordinates. Writing the norm as $\sqrt{\boldsymbol{x}\cdot\boldsymbol{x}}$ ties length to the inner product, which also gives angles and orthogonality. Scaling by $\alpha$ scales the length by $|\alpha|$ , so $-\boldsymbol{x}$ is as long as $\boldsymbol{x}$ .

The norm of a difference is the Euclidean distance $\|\boldsymbol{x}-\boldsymbol{y}\|$ $y ∥$ , and lengths satisfy the triangle inequality. The modulus of a complex number $a+b\mathrm{i}$ is the same length for the point $(a,b)$ . Math III later treats the Euclidean norm as one of several norms on $\mathbb{R}^n$ .

The norm is not the sum of the entries: $\|(1,1)\|=\sqrt{2}$ $2$ , not $2$ , and in $\mathbb{R}^n$ the vector $(1,\ldots,1)$ has norm $\sqrt{n}$ . Also $\|\boldsymbol{x}\|^2=\boldsymbol{x}\cdot\boldsymbol{x}$ $x$ , while $\|\boldsymbol{x}\|$ itself needs the square root.

Key Results¶

$\|\boldsymbol{x}\|\geq0$ $0$ , and $\|\boldsymbol{x}\|=0$ if and only if $\boldsymbol{x}=\boldsymbol{0}$ $0$ .
$\|\alpha \boldsymbol{x}\|=|\alpha|\|\boldsymbol{x}\|$ $∣ α ∣∥ x ∥$ .
If $\boldsymbol{x}\neq \boldsymbol{0}$ $0$ , then $\left\|\frac{1}{\|\boldsymbol{x}\|}\boldsymbol{x}\right\|=1$ $1$ .
The Euclidean norm is one example of a norm; other norms on $\mathbb{R}^n$ , such as the maximum norm, measure length differently.

Examples¶

If $\boldsymbol{x}=(3,4)$ $(3, 4)$ , then $\|\boldsymbol{x}\|=5$ $5$ .
The normalization of $(3,4)$ is $\left(\frac{3}{5},\frac{4}{5}\right)$ .

Prerequisites¶

Vector

Sources¶

Math II: Linear Algebra · Week 5
Math V: Advanced Linear Algebra · Week 5 (revisited)

Aliases: Euclidean length, 2-norm, length of a vector, L2 norm Keywords: norm, unit vector, normalization, normalized vector, normalised vector Symbols: alpha, element of, real numbers, square root

Last updated: 20 October 2026 03:02