2D Vector
|v| = √(x² + y²)
3D Vector
|v| = √(x² + y² + z²)
Physics: Calculating velocity, force, and acceleration magnitudes
Computer Graphics: Normalizing vectors, calculating distances
Engineering: Structural analysis, signal processing
Navigation: Distance and displacement calculations
The magnitude of a vector, also known as its length or norm, is a measure of how long the vector is. It represents the distance from the origin to the point defined by the vector's components. In physics and engineering, vector magnitude is essential for understanding quantities like speed (magnitude of velocity), force strength, and displacement.
The magnitude is always a non-negative scalar value, regardless of the direction of the vector. It is calculated using the Euclidean distance formula, which is derived from the Pythagorean theorem. For a 2D vector v = (x, y), the magnitude is √(x² + y²), and for a 3D vector v = (x, y, z), it extends to √(x² + y² + z²).
Non-negativity
|v| ≥ 0 for all vectors, and |v| = 0 only when v is the zero vector.
Scalar Multiplication
|cv| = |c| × |v| for any scalar c and vector v.
Triangle Inequality
|u + v| ≤ |u| + |v| for any vectors u and v.
Unit Vectors
A unit vector has magnitude 1 and is found by dividing v by |v|.
Disclaimer
Vector magnitude calculations follow standard Euclidean formulas. Results depend on correct component input and selected dimension. For complex applications, verify results with appropriate domain-specific tools.