Vector Magnitude Calculator Formula

Understand the math behind the vector magnitude calculator. Each variable explained with a worked example.

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Formulas Used

Magnitude

magnitude = mag

Unit X

unit_x = mag != 0 ? x / mag : 0

Unit Y

unit_y = mag != 0 ? y / mag : 0

Unit Z

unit_z = mag != 0 ? z / mag : 0

Direction Deg

direction_deg = atan(y / (x != 0 ? x : 0.0001)) * 180 / pi

Variables

VariableDescriptionDefault
xx component3
yy component4
zz component (0 for 2D)0
magDerived value= sqrt(pow(x,2) + pow(y,2) + pow(z,2))calculated

How It Works

Vector Magnitude

Formula

v= sqrt(x² + y² + z²)

For 2D vectors (z = 0): v= sqrt(x² + y²)

Unit Vector

A unit vector has magnitude 1 and points in the same direction:

v_hat = v /v= (x/v, y/v, z/v)

Direction (2D)

The angle from the positive x-axis: theta = arctan(y/x)

Worked Example

Find the magnitude and unit vector of (3, 4, 0).

x = 3y = 4z = 0
  1. 01|v| = √(9 + 16 + 0) = √25 = 5
  2. 02Unit vector = (3/5, 4/5, 0) = (0.6, 0.8, 0)
  3. 03Direction = arctan(4/3) ≈ 53.13°

Frequently Asked Questions

What is vector magnitude?

The magnitude (or length) of a vector is the distance from the origin to the point it represents. It is always non-negative.

What is a unit vector?

A unit vector has a magnitude of exactly 1. It indicates direction only. Any nonzero vector can be normalized to a unit vector by dividing by its magnitude.

How do I find the distance between two points?

Form a vector from one point to the other by subtracting coordinates, then find the magnitude. distance = √((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²).

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