Vector Norm and Distance Tool

Math Linear Algebra • Vectors and Basic Operations

Written by STEM Calculators Team Published February 21, 2026 Updated February 24, 2026

Compute vector norms \(\lVert\mathbf{V}\rVert_1\), \(\lVert\mathbf{V}\rVert_2\), \(\lVert\mathbf{V}\rVert_\infty\), and general \(\lVert\mathbf{V}\rVert_p=\left(\sum_i |v_i|^p\right)^{1/p}\). Also compute distance \(d_p(\mathbf{V},\mathbf{W})=\lVert\mathbf{V}-\mathbf{W}\rVert_p\).

Vector \(\mathbf{V}\) components Vector \(\mathbf{W}\) components

Norm / distance type

\(p\) (for custom)

Visualization

Display decimals

Show \( \lVert\mathbf{x}\rVert_p = 1\) “unit ball” (2D only)

If \(\mathbf{W}\) is blank, it defaults to \(\mathbf{0}\) (same dimension as \(\mathbf{V}\)). For \(n>2\), the 2D plot uses the first two components. “Play” animates \(p\) from 1 to 10 once and then stops automatically.

Ready

Norm geometry visualization

2D: drag to pan • wheel/trackpad to zoom • double-click to reset. 3D: drag to rotate • Shift+drag to pan • wheel/trackpad to zoom. Play shows a live overlay on the graph even in 3D.

\(\mathbf{V}\) \(\mathbf{W}\) \(\mathbf{V}-\mathbf{W}\) (distance vector) unit ball (2D)

Enter vectors and click “Calculate”.

Rate this calculator

0.0 /5 (0 ratings)

Be the first to rate.

Your rating

Name (optional) Review (optional)

You can update your rating any time.

Theory

Norms in \(\mathbb{R}^n\)

A norm assigns a nonnegative “length” to a vector \(\mathbf{v}=(v_1,\dots,v_n)\). Common norms include:

L1 (Manhattan): \[ \lVert \mathbf{v}\rVert_1=\sum_{i=1}^n |v_i|. \]
L2 (Euclidean): \[ \lVert \mathbf{v}\rVert_2=\sqrt{\sum_{i=1}^n v_i^2}. \]
L∞ (Chebyshev / max norm): \[ \lVert \mathbf{v}\rVert_\infty=\max_i |v_i|. \]
General Lp norm (p > 0): \[ \lVert \mathbf{v}\rVert_p=\left(\sum_{i=1}^n |v_i|^p\right)^{1/p}. \] (For norms in the strict sense, one typically uses \(p\ge 1\).)

Distance between vectors

A norm induces a distance (metric) by measuring the norm of the difference: \[ d_p(\mathbf{v},\mathbf{w})=\lVert \mathbf{v}-\mathbf{w}\rVert_p. \]

In 2D, these distances have geometric “unit circles”:

\(p=1\): diamond shape
\(p=2\): circle
\(p=\infty\): axis-aligned square
as \(p\) increases, the unit ball approaches the square (L∞)

Useful properties

Nonnegativity: \(\lVert \mathbf{v}\rVert \ge 0\) and equals \(0\) only for \(\mathbf{v}=\mathbf{0}\).
Scaling: \(\lVert c\mathbf{v}\rVert = |c|\,\lVert \mathbf{v}\rVert\).
Triangle inequality: \(\lVert \mathbf{v}+\mathbf{w}\rVert \le \lVert \mathbf{v}\rVert+\lVert \mathbf{w}\rVert\).

In applications, L1 often models “grid-like” movement, L2 models physical distance, and L∞ models “worst-case” deviation.

University note: infinite norms and infinite dimensions

The max norm \(\lVert\cdot\rVert_\infty\) also appears as a limit of Lp norms as \(p\to\infty\). More advanced topics study norms on function spaces (infinite-dimensional vector spaces), where \(\lVert f\rVert_p\) becomes an integral-based norm.

Rate this calculator

Related calculators