MathematicsVectorsA-Level
WJECAQASATIBAbiturAPBaccalauréat GénéralBachillerato

Vector magnitude Calculator

Find the magnitude of a 3D vector.

Use the free calculatorCheck the variablesOpen the advanced solver
Open Advanced Calculator
This is the free calculator preview. Advanced walkthroughs stay in the app.
Result
Ready
Magnitude

Formula first

Overview

Vector magnitude, also known as the Euclidean norm, represents the total length or distance of a vector from its origin to its tip in a three-dimensional Cartesian coordinate system. It is calculated by taking the square root of the sum of the squares of the vector's orthogonal components, effectively applying the Pythagorean theorem to 3D space.

Symbols

Variables

a_x = x-component, a_y = y-component, a_z = z-component, |\mathbf{a}| = Magnitude

x-component
y-component
z-component
Magnitude

Apply it well

When To Use

When to use: Apply this formula whenever you need to convert vector components into a single scalar value representing size, strength, or distance. It is used in scenarios where direction is known or given via components and only the total magnitude is required for further calculation.

Why it matters: This calculation is foundational in physics for determining the strength of force fields, the speed of an object from velocity components, and the distance between points in space. In engineering and computer science, it is essential for normalizing vectors to create unit vectors used in lighting and motion simulations.

Avoid these traps

Common Mistakes

  • Adding components then rooting.
  • Sign errors cancelling squares.

One free problem

Practice Problem

A displacement vector has components of 3 meters along the x-axis, 4 meters along the y-axis, and 12 meters along the z-axis. Calculate the total magnitude of this displacement.

x-component3
y-component4
z-component12

Solve for:

Hint: Square each component, add them together, and then find the square root of the total.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Halliday, Resnick, Walker, Fundamentals of Physics
  2. Wikipedia: Euclidean vector
  3. Stewart, Calculus: Early Transcendentals
  4. Halliday, Resnick, Walker, Fundamentals of Physics, 11th Edition
  5. Halliday, Resnick, Walker Fundamentals of Physics
  6. Stewart Calculus: Early Transcendentals
  7. Wikipedia article 'Euclidean vector'
  8. Wikipedia article 'Norm (mathematics)'