Unit Vector Calculator – Normalize a Vector
Enter a vector to find the unit vector in the same direction.
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
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)