23. Cross Product Magnitude
The norm of the cross product is related to the angle between the two vectors:
where is the angle between and .
This is the signed area of the parallelogram spanned by the two vectors:
23.1. Geometric meaning
- when are parallel (or anti-parallel) — parallelogram degenerates to a line segment, area
- when are perpendicular — maximum area
23.2. Connection to the determinant
The magnitude is the absolute value of the determinant with as two columns and any orthonormal third column — equivalently the area of the projection of the parallelogram into the plane perpendicular to that third axis.
For a triple of vectors, the triple product gives the signed volume of the parallelepiped — the determinant with those vectors as columns.
23.3. Connections
- Cross Product — the operation itself
- Dot vs Cross — side-by-side
- Determinant — signed parallelogram / parallelepiped volume
- Angles Between Vectors — recovering from the formula