Vector Norm — Definition, Formula & Examples

A vector norm is a function that assigns a non-negative length or size to a vector. The most common type is the Euclidean norm, which gives the straight-line distance from the origin to the tip of the vector.

A norm on a vector space $V$ over $\mathbb{R}$ (or $\mathbb{C}$ ) is a function $\|\cdot\|: V \to \mathbb{R}$ satisfying three axioms for all vectors $\mathbf{u}, \mathbf{v} \in V$ and all scalars $\alpha$ : (1) $\|\mathbf{v}\| \geq 0$ , with equality if and only if $\mathbf{v} = \mathbf{0}$ (positive definiteness); (2) $\|\alpha\mathbf{v}\| = |\alpha|\,\|\mathbf{v}\|$ (absolute homogeneity); (3) $\|\mathbf{u} + \mathbf{v}\| \leq \|\mathbf{u}\| + \|\mathbf{v}\|$ (triangle inequality).

Key Formula

\|\mathbf{v}\|_p = \left(\sum_{i=1}^{n} |v_i|^p\right)^{1/p}

Where:

$\mathbf{v}$ = A vector with components $v_1, v_2, \ldots, v_n$
$n$ = The number of components (dimension of the vector)
$p$ = A positive real number specifying the norm type; $p = 2$ gives the Euclidean norm

How It Works

To compute the Euclidean norm (also called the

L^2

norm), square each component of the vector, sum the squares, and take the square root. Other norms exist: the

L^1

norm sums absolute values, and the

L^\infty

norm takes the largest absolute value among the components. The choice of norm depends on the application, but the Euclidean norm is the default in most introductory courses. Any function satisfying the three norm axioms qualifies as a valid norm.

Worked Example

Problem: Find the Euclidean norm of the vector

\mathbf{v} = (3, -4, 12)

Square each component: Compute the square of each entry.

3^2 = 9, \quad (-4)^2 = 16, \quad 12^2 = 144

Sum the squares: Add the results together.

9 + 16 + 144 = 169

Take the square root: Apply the square root to get the norm.

\|\mathbf{v}\| = \sqrt{169} = 13

Answer:

\|\mathbf{v}\| = 13

Why It Matters

Vector norms are essential in machine learning for regularization (e.g.,

L^1

and

L^2

penalties in regression) and in numerical analysis for measuring approximation errors. In physics and engineering, the Euclidean norm directly represents quantities like force magnitude, velocity, and displacement.

Common Mistakes

Mistake: Forgetting to take the square root after summing the squared components.

Correction: The Euclidean norm requires the square root of the sum of squares:

\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}

. Without it, you have the squared norm

\|\mathbf{v}\|^2

, which is a different quantity.