What formula is used to find the magnitude of a vector?

The Euclidean norm (magnitude) is |v| = sqrt(x² + y² + z²) in three dimensions, and generalises to |v| = sqrt(x₁² + x₂² + ... + xₙ²) for an n-dimensional vector. It comes directly from the Pythagorean theorem applied repeatedly: the magnitude is the straight-line distance from the origin to the tip of the vector.

What is a unit vector and how is it calculated?

A unit vector v̂ is a vector that points in the same direction as v but has magnitude exactly 1. It is computed by dividing every component by the magnitude: v̂ = v / |v|. Unit vectors are used to represent direction without scale — in physics, engineering and computer graphics they appear constantly in normal vectors, rotation axes and lighting calculations.

What are direction cosines and direction angles?

For a 3-D vector v = (x, y, z) with magnitude |v|, the direction cosines are cos α = x/|v|, cos β = y/|v|, cos γ = z/|v|. The angles α, β, γ are called direction angles — they are the angles the vector makes with the positive x-, y- and z-axes respectively. A key identity is cos²α + cos²β + cos²γ = 1, which the calculator verifies for you.

How do I find the magnitude of a vector in two dimensions?

In 2-D, |v| = sqrt(x² + y²). This is the classic Pythagorean theorem: x and y are the legs of a right triangle, and |v| is the hypotenuse. For example, v = (3, 4) gives |v| = sqrt(9 + 16) = sqrt(25) = 5.

Can I solve for a missing component if I know the magnitude?

Yes — use the Solve tab. Given |v|, x and y, the missing z component satisfies z² = |v|² - x² - y², so z = ±sqrt(|v|² - x² - y²). There are generally two solutions (positive and negative) because a vector pointing above and one pointing below the xy-plane can have the same magnitude, x and y values.

Why does the calculator use Math.hypot instead of manually squaring and summing?

Math.hypot uses a numerically stable algorithm that avoids overflow (components near the floating-point maximum) and underflow (very small components). Manually computing sqrt(x² + y² + z²) can lose precision or even produce Infinity for large inputs — hypot rescales internally to keep the result accurate across the full numeric range.

Vector Magnitude Calculator

A vector magnitude calculator that finds the Euclidean length of any vector — 2-D, 3-D, or arbitrary n-dimensional — shows the full working, computes the unit vector, and (in 3-D) lists the direction cosine angles. A dedicated solve-for-component mode lets you work backwards: supply the magnitude and two known components, and the calculator returns the missing third. Everything runs locally in your browser; no numbers leave your device.

How it works

The fundamental formula is the Euclidean norm, a direct extension of the Pythagorean theorem to higher dimensions:

|v| = sqrt(x₁² + x₂² + … + xₙ²)

In two dimensions this is |v| = sqrt(x² + y²) — the hypotenuse of the right triangle formed by the components. In three dimensions it extends to |v| = sqrt(x² + y² + z²). The pattern continues identically for any number of dimensions, which is why the n-D tab accepts an unlimited comma-separated list of components.

Unit vector

Once you have the magnitude, the unit vector v̂ is obtained by dividing every component by |v|:

v̂ = v / |v| = (x/|v|, y/|v|, z/|v|)

By construction |v̂| = 1. Unit vectors encode direction only and appear throughout physics (force direction, surface normals, polarisation vectors), computer graphics (lighting, rotation axes), and machine learning (cosine similarity).

Direction angles in 3-D

For a 3-D vector the direction cosines are the cosines of the angles the vector makes with each coordinate axis:

cos α = x / |v|, cos β = y / |v|, cos γ = z / |v|

A fundamental identity — sometimes called the direction cosine identity — guarantees:

cos²α + cos²β + cos²γ = 1

The calculator verifies this identity numerically so you can see the constraint holding in real time.

Solving for a missing component

Given |v|, x, and y, the z component satisfies:

z = ± sqrt(|v|² − x² − y²)

This has two real solutions (±) as long as x² + y² does not exceed |v|². If it does, there is no real z that could produce the given magnitude with those x and y components, and the calculator explains why.

Worked example

Suppose you are working in 3-D and your vector is v = (2, 6, 3).

Sum of squares: 2² + 6² + 3² = 4 + 36 + 9 = 49
Magnitude: |v| = √49 = 7
Unit vector: v̂ = (2/7, 6/7, 3/7) ≈ (0.285714, 0.857143, 0.428571)
Direction angles:
- α = arccos(2/7) ≈ 73.4°
- β = arccos(6/7) ≈ 31.0°
- γ = arccos(3/7) ≈ 64.6°
Identity check: cos²(73.4°) + cos²(31.0°) + cos²(64.6°) ≈ 0.0816 + 0.7347 + 0.1837 = 1.000 ✓

For the n-D tab, entering 1, 2, 3, 4, 5 gives |v| = √(1 + 4 + 9 + 16 + 25) = √55 ≈ 7.416, and the five-component unit vector is approximately (0.1348, 0.2697, 0.4045, 0.5394, 0.6742).

Vector	Magnitude	Unit vector (3 d.p.)
(3, 4)	5	(0.600, 0.800)
(1, 1, 1)	1.732	(0.577, 0.577, 0.577)
(2, 6, 3)	7	(0.286, 0.857, 0.429)
(0, 0, 5)	5	(0, 0, 1)
(1, 2, 3, 4, 5)	7.416	(0.135, 0.270, 0.404, 0.539, 0.674)

All results are computed to six significant figures; the internal implementation uses Math.hypot for numerical stability across the full floating-point range.