Vector Magnitude Calculator

Vector Magnitude Calculator – From 2D Length To N-Dimensional Norms

A vector is an ordered list of numbers that represent a direction and a size in space. The magnitude of a vector, often written |v|, is the Euclidean length of that vector: the straight-line distance from the origin to the point represented by the vector. This Vector Magnitude Calculator on MyTimeCalculator lets you compute vector length in 2D, 3D and N dimensions, build unit vectors, measure distance between vectors and compare different vector norms.

Throughout this article we write a vector as v = (x₁, x₂, …, xₙ). The same formulas apply whether n = 2, 3 or any positive integer.

Magnitude Of A Vector – Core Formulas

For a vector v = (x₁, x₂, …, xₙ), the Euclidean magnitude is defined by

|v| = √(x₁² + x₂² + … + xₙ²)

This is a direct extension of the Pythagorean theorem to higher dimensions.

Important special cases include:

• 2D vector v = (x, y): |v| = √(x² + y²)
• 3D vector v = (x, y, z): |v| = √(x² + y² + z²)

In all cases, the squared magnitude is the sum of squared components:

|v|² = x₁² + x₂² + … + xₙ²

Unit Vector – Direction With Magnitude 1

A unit vector captures direction only. Given a nonzero vector v, its unit vector v̂ is defined as

v̂ = v / |v|

Written component-wise, if v = (x₁, …, xₙ) and |v| ≠ 0, then

v̂ = (x₁ / |v|, x₂ / |v|, …, xₙ / |v|)

The magnitude of v̂ is always 1:

|v̂| = 1

Unit vectors are widely used to represent directions, normals to surfaces and axis directions in 2D and 3D geometry.

Distance Between Two Vectors

Suppose two vectors A and B have the same dimension and are written as

• A = (a₁, a₂, …, aₙ)
• B = (b₁, b₂, …, bₙ)

The difference vector is A − B = (a₁ − b₁, a₂ − b₂, …, aₙ − bₙ). The Euclidean distance between A and B is the magnitude of this difference:

d₂(A, B) = |A − B| = √((a₁ − b₁)² + (a₂ − b₂)² + … + (aₙ − bₙ)²)

In 2D this is the familiar distance formula between two points; in higher dimensions it generalizes the same idea.

Vector Norms – L1, L2 And L∞

Magnitude is one example of a vector norm. A norm is a function that assigns a non-negative length to each vector and satisfies certain properties. Three common norms are:

L1 norm (Manhattan norm)

For v = (x₁, …, xₙ), the L1 norm is

||v||₁ = |x₁| + |x₂| + … + |xₙ|

This measures distance as if you could move only along coordinate axes, similar to moving along a grid of city streets.

L2 norm (Euclidean norm)

The L2 norm is the usual notion of magnitude:

||v||₂ = √(x₁² + x₂² + … + xₙ²)

This is the length returned by the 2D, 3D and N-dimensional magnitude modes of the calculator.

L∞ norm (maximum norm)

The L∞ norm is defined as the maximum absolute component:

||v||∞ = max(|x₁|, |x₂|, …, |xₙ|)

This norm measures how large the largest component is and is often used in numerical analysis and error bounds.

How The Vector Magnitude Calculator Works

The calculator is divided into several modes that correspond to these formulas.

Mode 1: 2D Vector Magnitude

In the 2D tab you enter components x and y. The calculator computes

• Squared magnitude: |v|² = x² + y²
• Magnitude: |v| = √(x² + y²)
• Unit vector v̂ = (x / |v|, y / |v|) if |v| ≠ 0

A short table shows each component and its squared value so you can see how they add up to |v|².

Mode 2: 3D Vector Magnitude

In the 3D tab you enter x, y and z. The formulas extend to

• Squared magnitude: |v|² = x² + y² + z²
• Magnitude: |v| = √(x² + y² + z²)
• Unit vector v̂ = (x / |v|, y / |v|, z / |v|) when |v| is nonzero

This is the standard formula for vector length in 3D coordinate geometry and physics.

Mode 3: N-Dimensional Magnitude

The N-dimensional tab lets you work in any dimension. You can add as many components as you like. If v = (x₁, …, xₙ), the calculator evaluates

• Squared magnitude: |v|² = Σ xᵢ²
• Magnitude: |v| = √(Σ xᵢ²)
• L1 norm: ||v||₁ = Σ |xᵢ|
• L∞ norm: ||v||∞ = max |xᵢ|

All components appear in a table with their index, value and squared value.

Mode 4: Unit Vector Calculator

In the unit vector tab you enter the components of any nonzero vector. The calculator computes

• Magnitude |v| = √(Σ xᵢ²)
• Unit vector components: xᵢ / |v| for each i
• Magnitude of the unit vector, which should equal 1 up to rounding

If you enter the zero vector (all components equal to 0), a unit vector cannot be defined because there is no direction to normalize.

Mode 5: Distance Between Two Vectors

For two vectors A = (a₁, …, aₙ) and B = (b₁, …, bₙ) in the same dimension, the calculator computes three distances:

• L2 distance: d₂(A, B) = √(Σ(aᵢ − bᵢ)²)
• L1 distance: d₁(A, B) = Σ|aᵢ − bᵢ|
• L∞ distance: d∞(A, B) = max|aᵢ − bᵢ|

A table shows each component of A and B, the difference aᵢ − bᵢ and the squared difference (aᵢ − bᵢ)² so you can see how the final distance is built up.

Mode 6: Vector Norms

The norms tab focuses on a single vector v = (x₁, …, xₙ) and calculates all three norms together:

• ||v||₁ = Σ|xᵢ|
• ||v||₂ = √(Σxᵢ²)
• ||v||∞ = max|xᵢ|

A short summary table reminds you of each definition and shows how they differ conceptually.

Practical Tips For Working With Vector Magnitudes

• Use 2D and 3D modes for quick geometry and physics problems in the plane or in space.
• Use the N-dimensional mode when you work with data vectors, feature vectors or any situation with more than 3 components.
• Use the unit vector mode when you need a direction with length 1, such as when defining a direction vector or a normal.
• Use the distance mode to compare two vectors or points and see how far apart they are under different norms.
• Compare L1, L2 and L∞ norms in the norms tab to understand how different norm choices emphasize different aspects of a vector.

This Vector Magnitude Calculator is designed for learning, homework, engineering sketches and quick checks. For large-scale numerical work or high-dimensional data analysis, combine these formulas with specialized numerical software.