Vector Magnitude Calculator

Created by Wojciech Sas, PhD
Reviewed by Bogna Szyk and Jack Bowater
Last updated: Oct 31, 2022

This vector magnitude calculator is a simple tool that helps you estimate magnitude based on vector components. In the text, you'll learn how to find the magnitude of a vector and get used to the general magnitude of a vector formula. At the end of the text, you can read about some physical quantities which are vectors, and what their magnitudes tell us.

How to find the magnitude of a vector?

We can define a vector can as an ordered collection of numbers. The number of vector components depends on the dimension of the space. In practice, we usually deal with 3-dimensional vectors with three distinct components. In Cartesian coordinates, we can use the values of the x, y, and z components.

🙋 Feel free to use Omni's three-dimensional distance calculator to save time when dealing with such math problems!

When we switch to spherical coordinates, it's convenient to use the values of two angles, θ & φ, and the magnitude, which is the length of a vector in its purest meaning. In other words, it's the three-dimensional distance between the initial and end point of a vector.

The components of a vector can be complex numbers as well.

The magnitude of a vector formula

The magnitude of a vector |V| can be estimated in numerous ways, depending on the dimensionality of the vector space. We have:

  • |V| = √(x² + y²) in 2-d space;
  • |V| = √(x² + y² + z²) in 3-d space;
  • |V| = √(x² + y² + z² + t²) in 4-d space;
  • |V| = √(x² + y² + z² + t² + w²) in 5-d space, and so on…

As you can see in the formula for the magnitude of a vector, magnitude is the square root of the sum of vector components to the second power in all cases. In this vector magnitude calculator, you can set the dimensionality of your vector so that the correct formula is chosen. As a result, the magnitude's value is always positive, which is why we can measure it in any experiment dealing with vector quantities.

We can also calculate the magnitude of a vector as the square root of the dot product of the vector with itself:

|V| = √(V·V)

By definition, the magnitude of a unit vector is 1. Our unit vector calculator helps you learn how to normalize any vector.

Matrices can also have magnitudes, or at least, they have norms that indicate the magnitude of stretching a vector it's multiplied with.

How to use the vector magnitude calculator?

Let's take a look at this computational example to learn how to find the magnitude of a vector in 4-dimensional space. The components of the vector are x = 3, y = -1, z = 2, t = -3.

  1. Estimate the squares of each vector component: x² = 9, y² = 1, z² = 4, t² = 9.
  2. Add them all together: x² + y² + z² + t² = 9 + 1 + 4 + 9 = 23.
  3. Work out the magnitude of the vector as a square root of these values: |V| = √23 = 4.796.

Examples of vectors in real life

A lot of physical quantities (e.g., force, acceleration, and velocity) are vectors. In these cases, the magnitude of a vector is the absolute value of the measured quantity, like how speed is the magnitude of velocity.

🙋 Omni's magnitude of acceleration calculator is an excellent tool for understanding the concept of vectors' magnitudes in physics!

You can create other, more complex ones, by evaluating the cross product of two vectors. Torque is a cross product of distance and force vectors.

Wojciech Sas, PhD
3-D vector
Input coordinates
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