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°

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