To normalize a vector, you calculate its magnitude (length) and then divide each component of the vector by that magnitude. This results in a unit vector, which has the same direction as the original vector but a magnitude of 1.
Here's a breakdown of the process:
1. Calculate the Magnitude (Length) of the Vector
The magnitude of a vector v = (x, y, z) is calculated using the Euclidean norm:
||v|| = √(x² + y² + z²)
For a 2D vector v = (x, y):
||v|| = √(x² + y²)
Example:
Let's say we have a 2D vector v = (3, 4).
||v|| = √(3² + 4²) = √(9 + 16) = √25 = 5
2. Divide Each Component by the Magnitude
Once you have the magnitude, divide each component of the original vector by this value.
Normalized vector v̂ = (v / ||v||)
So, v̂ = (x / ||v||, y / ||v||, z / ||v||) for a 3D vector.
Example (continuing from above):
v = (3, 4), ||v|| = 5
v̂ = (3/5, 4/5) = (0.6, 0.8)
This normalized vector v̂ = (0.6, 0.8) has a magnitude of 1:
||v̂|| = √(0.6² + 0.8²) = √(0.36 + 0.64) = √1 = 1
Summary:
- Find the magnitude (length) of the vector.
- Divide each component (x, y, z, etc.) of the original vector by its magnitude. The resulting vector is the normalized vector.
The normalized vector points in the same direction as the original vector, but its length is exactly 1. This is useful in many applications, such as:
- Direction Vectors: Representing a direction without being influenced by magnitude.
- Lighting Calculations in Computer Graphics: Ensuring consistent lighting effects.
- Machine Learning: Feature scaling and improving algorithm performance.
