# Unit Vector Calculator

By Bogna Szyk and Hanna Pamuła, PhD candidate
Last updated: Aug 21, 2021

This unit vector calculator will help you transform any vector into a vector of length 1 without changing its direction. If you want to know how to calculate a unit vector's components, look no further! You can obtain the result by dividing the components of any arbitrary vector by its magnitude. Don't worry if you don't know how to find a vector's magnitude, though. This article will give you a step-by-step explanation.

## What is a unit vector?

A unit vector is a vector of length equal to 1.

When the unit vector is used to describe a spatial direction, it can be called a direction vector. In a Cartesian coordinate system, the three unit vectors that form the basis of the 3D space are:

• (1, 0, 0) - describes the x-direction
• (0, 1, 0) - describes the y-direction
• (0, 0, 1) - describes the z-direction

Every vector in a 3D space is equal to a sum of unit vectors.

## Unit vector formula

If you are given an arbitrary vector, it is possible to calculate what is the unit vector along the same direction. To do that, you have to apply the following formula:

`û = u / |u|`

where:

• û is the unit vector,
• u is an arbitrary vector in the form (x, y, z), and
• |u| is the magnitude of the vector u.

You can calculate the magnitude of a vector using our distance calculator or simply by the equation

`|u| = √(x² + y² + z²)`

Calculating the magnitude of a vector is also a useful skill for finding the midpoint of a segment.

The unit vector is a useful concept in describing linear transformations. For example, the matrix norm describes how much a unit vector is stretched when multiplied with a matrix.

## How to calculate the unit vector

Let's consider an example of a vector u = (8, -3, 5). To calculate the unit vector in the same direction, you have to follow these steps:

1. Write down the x, y and z components of the vector. In this case, x₁ = 8, y₁ = -3 and z₁ = 5.

2. Calculate the magnitude of the vector u:

`|u| = √(x₁² + y₁² + z₁²)`

`|u| = √(8² + (-3)² + 5²)`

`|u| = √(64 + 9 + 25)`

`|u| = √98`

`|u| = 9.9`

1. Now that you know the magnitude of the vector u, you probably want to know how to calculate the unit vector. All you have to do is divide each of the initial vector's components by |u|.

`x₂ = x₁ / |u| = 8 / 9.9 = 0.8081`

`y₂ = y₁ / |u| = -3 / 9.9 = -0.3031`

`z₂ = z₁ / |u| = 5 / 9.9 = 0.5051`

1. Now, write these results in a vector form to find the vector û = (0.8081, -0.3031, 0.5051).

2. You can check whether the result is correct. If it is, the magnitude of your unit vector should be equal to 1.

Bogna Szyk and Hanna Pamuła, PhD candidate
I want to...
normalize vector
Find a unit vector with the same direction as your given vector.
Dimension
2D Original vector
x
y
Direction angle θ
deg
Magnitude
💡 The x and y components are squared in the formula for the vector magnitude. Because of this, you can use either negative or positive values and still get the same magnitude.
Unit vector
x
y
Magnitude
1
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