Unit Vector Calculator – Normalize a Vector
Enter a vector to find the unit vector in the same direction.
Results: Unit Vector
Magnitude
Steps to Solve:
Use the Unit Vector Formula
â = a / |a|
Step One: Solve the Magnitude
|a| = x² + y²
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
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
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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|
So, 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
zû = 8 / √98 = 0.808
So, the unit vector û is (0.303, 0.505, 0.808).
û = (0.303, 0.505, 0.808)