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Calculation Machine

Vector Magnitude Calculator

Length, unit vector and direction angles in 2D or 3D.

Reviewed by the Calculation Machine team · How we check our maths

Your values

ⓘ Live results

Results
Magnitude
13
Unit vector
(0.2308, 0.3077, 0.9231)
Angle from x-axis
76.66°
Angle from y-axis
72.08°
Angle from z-axis
22.62°
2D angle in the xy-plane
53.13°

Quick answer

A vector's magnitude is the square root of the sum of its squared components: |v| = √(x² + y² + z²).

The formula

|v| = √(x² + y² + z²)

Worked example

Worked example

Using the values the calculator loads with.

Inputs

  • x component3
  • y component4
  • z component12

Result

  • Magnitude13
  • Unit vector(0.2308, 0.3077, 0.9231)
  • Angle from x-axis76.66°
  • Angle from y-axis72.08°
  • Angle from z-axis22.62°
  • 2D angle in the xy-plane53.13°

How to use the Vector Magnitude Calculator

  1. 1Enter the x and y components.
  2. 2Leave z as 0 for a 2D vector.
  3. 3Use the unit vector when you need direction without magnitude.

Frequently asked questions

What is a unit vector for?

It carries pure direction — multiply it by any length to build a vector pointing the same way.

Are direction cosines the same thing?

Yes, the unit vector's components are the cosines of the angles with each axis, and their squares sum to 1.

About this calculator