Unit Vector Calculator

📘 Formula

To find a unit vector $\mathbf{\hat{u}}$ given a vector $\mathbf{v}$, divide the vector by its magnitude $|\mathbf{v}|$.

🎓 Step-by-Step Example

Find the unit vector for v = (3, 4).

1. Calculate Magnitude: |v| = √(3² + 4²) = √(9 + 16) = √25 = 5.
2. Divide Components: u = (3/5, 4/5).
3. Result: u = (0.6, 0.8).

Verify by checking the length of u: √(0.6² + 0.8²) = √(0.36 + 0.64) = √1 = 1.

🚀 What is "Normalization"?

The process of converting a vector to a unit vector is called Normalization. It strips away the "strength" or "speed" of the vector, leaving only its direction.

This is crucial in computer graphics for calculating lighting. The angle at which light hits a surface depends only on the direction of the light ray and the surface normal, not how "long" those mathematical lines are.

🧠 The Zero Vector Anomaly

The vector (0, 0) has a magnitude of 0.

Since you cannot divide by zero, the unit vector for the zero vector is undefined. It has no direction. In programming (like GLSL), normalizing a zero vector often results in NaN (Not a Number) or errors unless handled specifically.