Unit Vector Calculator – Normalize a Vector

Enter a vector to find the unit vector in the same direction.

Vector Coordinates
Vector Coordinates

Unit Vector:

 

Magnitude

 

Steps to Solve

Use the Unit Vector Formula

â = a / |a|

Step One: Solve the Magnitude

|a| = x² + y² + z²

Substitute Values and Solve

Enter vector coordinates above to see the solution here

Step Two: Divide by the Magnitude

Divide each vector component by the magnitude.

Substitute Values and Solve

Enter vector coordinates above to see the solution here

Learn how we calculated this below


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How to Find a Unit Vector

A unit vector is a vector with a length, or magnitude, of 1. You can scale a vector to a unit vector by reducing its length to 1 without changing its direction.

This is often referred to as vector normalization.

Unit Vector Formula

To normalize a vector to a unit vector, use the following formula:

û = u / |u|

Thus, the unit vector û of vector u is equal to each component of vector u divided by its magnitude |u|.

How to Use the Unit Vector Formula

The first step to scale a vector to a unit vector is to find the vector’s magnitude. You can use the magnitude formula to find it.

|u|= x² + y² + z²

The magnitude |u| of vector u is equal to the square root of the sum of the square of each of the vector’s components x, y, and z.

Then, divide each component of vector u by the magnitude |u|. The resulting components form the unit vector.

For example, given a vector (3, 5, 8), let’s find the unit vector.

Start by solving the magnitude.

|u|= 3² + 5² + 8²
|u|= 9 + 25 + 64
|u|= 98

Then, divide each vector coordinate by the magnitude 98.

xû = 3 / √98 = 0.303
yû = 5 / √98 = 0.505
zû = 8 / √98 = 0.808

So, the unit vector û is (0.303, 0.505, 0.808).

û = (0.303, 0.505, 0.808)