# Vector Magnitude Calculator

By Wojciech Sas, PhD candidate
Last updated: Aug 21, 2021

This vector magnitude calculator is a simple tool which 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?

A vector can be defined 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.

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, and that's why we can measure it in any experiment where we're dealing with vector quantities.

The magnitude of a vector can be also calculated 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`. 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.

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 candidate
Space
3-D vector
Input coordinates
x
y
z
Magnitude
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