Enter any vector into the normalized vector calculator. The calculator will normalize this vector and display the unit vector.

Normalizing a Vector Formula

The following formula is used to normalize a vector. 

u = U / |U|
|U| = Square Root ( X^2 + Y^2+Z^2)
  • Where U is the original vector
  • |U| is the magnitude of the vector
  • u is the unit vector

Normalize Vector Definition

A normalized vector is a vector that has been scaled to have a magnitude of 1 while preserving its direction. It is obtained by dividing each component of the vector by its magnitude.

Normalizing a vector is important in various fields, including mathematics, physics, computer science, and data analysis. By normalizing vectors, we bring them to a standardized scale, which allows for easier comparison and analysis. Normalized vectors have several useful properties that make them important in these fields.

Normalized vectors simplify calculations involving vector quantities. When vectors are normalized, their magnitudes become equal to 1, which eliminates the need to consider their original scale. This simplifies mathematical operations such as dot products, cross products, and projections.

How to normalize a vector?

How to normalize a vector?

  1. First, calculate the magnitude of the original vector

    Using the formula above, calculate the magnitude of the original vector.

  2. Next, divide each component of the vector by the magnitude.

    For example, for a vector x,y,z, divide x by the magnitude, y by the magnitude, and z by the magnitude. The results of those divisions are your unit vector values.

Example Problem:

In the following example, a vector of (5,6,10) is given.

First, the magnitude of the vector must bed calculated. Using the formula above:

|U| = sqrt( 5^2 + 6^2+10^2)

|U| = 12. 688

Next, divide each individual component of the vector by the magnitude to normalize the vector.

X = 5 / 12.688 = .394

Y = 6 / 12.688 = .472

Z = 10 / 12.688 = .788

So the final normalized vector would be (.394,.472,.788).

FAQ

What is normalizing a vector?

Normalizing a vector is the process of turning a vector into its unit vector. This process involves dividing a vector by its magnitude. The result is a vector with the same direction, but with a magnitude of 1.

Why would you normalize a vector?

Normalizing a vector can simply problems. For example, if you want to multiply two vectors A and B, you can actually multiply their unit vectors to get the direction, then multiply that answer by the magnitudes to get the resulting vector of A * B.