length of vector calculator

The length of a vector (also called vector magnitude or norm) tells you how big a vector is, regardless of direction. If you are solving geometry problems, building a game engine, studying physics, or working through linear algebra homework, this is one of the most common vector calculations you will use.

What is vector length?

A vector can be written as a list of components, such as (x, y) in 2D or (x, y, z) in 3D. The length of that vector is the distance from the origin to the point represented by the vector. Geometrically, it is an extension of the Pythagorean theorem.

Formulas by dimension

2D vectors

If v = (x, y), then:

||v|| = √(x² + y²)

3D vectors

If v = (x, y, z), then:

||v|| = √(x² + y² + z²)

n-dimensional vectors

The same idea scales to any number of dimensions. Just square each component, add them all, then take the square root of the sum.

How to use this length of vector calculator

• Type your vector components in a single line.
• Use commas, spaces, or semicolons as separators.
• Click Calculate Length to see the magnitude and full calculation steps.
• Use Clear to reset inputs and start over.

Worked examples

Example 1: vector (3, 4)

Compute:

||v|| = √(3² + 4²) = √(9 + 16) = √25 = 5

Example 2: vector (2, -1, 2)

Compute:

||v|| = √(2² + (-1)² + 2²) = √(4 + 1 + 4) = √9 = 3

Example 3: vector (1, 2, 2, 1)

Compute:

||v|| = √(1 + 4 + 4 + 1) = √10 ≈ 3.1623

Where vector magnitude is used

• Physics: Speed, force, and acceleration magnitudes.
• Computer graphics: Normalizing direction vectors for lighting and movement.
• Machine learning: Measuring distances and vector norms in feature space.
• Navigation and robotics: Path planning and displacement calculations.

Common mistakes to avoid

• Forgetting to square negative values correctly.
• Adding components directly without squaring first.
• Mixing separators and accidental text input (like units in the same field).
• Rounding too early during intermediate steps.

Quick FAQ

Is vector length ever negative?

No. Magnitude is always zero or positive.

What is the length of the zero vector?

It is exactly 0, because all components are zero.

Is this the same as Euclidean norm?

Yes. This calculator uses the Euclidean (L2) norm formula.

Can I enter decimals?

Absolutely. Values like 1.5, -2.75, and 0.001 are all supported.

Final thoughts

A reliable length of vector calculator saves time and reduces arithmetic mistakes, especially with higher-dimensional vectors. Use it to verify homework steps, test algorithms, or quickly solve practical engineering and data problems.