Understanding Vector Magnitude
The magnitude of a vector, also called its length or norm, measures the distance from the origin to the point represented by the vector. It is always a non-negative scalar value and is fundamental in physics, engineering, and computer graphics.
Magnitude Formulas
2D Vector Magnitude
For a vector v = (x, y), the magnitude is the hypotenuse of a right triangle.
|v| = sqrt(x² + y²)
3D Vector Magnitude
Extended to three dimensions with an additional z-component.
|v| = sqrt(x² + y² + z²)
Unit Vector
A unit vector has magnitude 1 and preserves only direction.
u = v / |v|
Applications of Vector Magnitude
Vector magnitude is used extensively in physics to represent speed (magnitude of velocity), force strength, displacement distance, and electric field intensity. In computer graphics, it is essential for normalizing vectors, calculating distances between points, and performing lighting calculations.
Key Properties
- The magnitude is always non-negative: |v| >= 0.
- |v| = 0 if and only if v is the zero vector.
- Scaling: |cv| = |c| * |v| for any scalar c.
- Triangle inequality: |a + b| <= |a| + |b|.