Vector Magnitude Calculator — Formula
## Vector Magnitude
### Formula
**|v| = sqrt(x² + y² + z²)**
For 2D vectors (z = 0): **|v| = sqrt(x² + y²)**
### Unit Vector
A unit vector has magnitude 1 and points in the same direction:
**v_hat = v / |v| = (x/|v|, y/|v|, z/|v|)**
### Direction (2D)
The angle from the positive x-axis: **theta = arctan(y/x)**
### Formula
**|v| = sqrt(x² + y² + z²)**
For 2D vectors (z = 0): **|v| = sqrt(x² + y²)**
### Unit Vector
A unit vector has magnitude 1 and points in the same direction:
**v_hat = v / |v| = (x/|v|, y/|v|, z/|v|)**
### Direction (2D)
The angle from the positive x-axis: **theta = arctan(y/x)**
Esempio Risolto
Find the magnitude and unit vector of (3, 4, 0).
- |v| = √(9 + 16 + 0) = √25 = 5
- Unit vector = (3/5, 4/5, 0) = (0.6, 0.8, 0)
- Direction = arctan(4/3) ≈ 53.13°