where the numerator is the determinant
and in the denominator the n-fold product
of the magnitudes ||vi|| of the vectors equals the hypervolume of the n-dimensional hyperrectangle, with edges equal to the magnitudes of the vectors ||v1||, ||v2||, ... ||vn|| (not the vectors themselves). Also see Ericksson.
The parallelotope is like a "squashed hyperrectangle", so it has less hypervolume than the hyperrectangle, meaning (see image for the 3d case):
as for the ordinary sine, with either bound only being reached in case all vectors are mutually orthogonal.
In case n = 2, the polar sine is the ordinary sine of the angle between the two vectors.
If the dimension of the space is more than n, then the polar sine is non-negative; otherwise it changes signs whenever two of the vectors vj and vk are interchanged - due to the antisymmetry of row-exchanging in the determinant:
- Invariance under scalar multiplication of vectors
The absolute value of the polar sine does not change if all of the vectors v1, ..., vn are multiplied by positive constants ci, due to factorization:
- Trigonometric functions
- List of trigonometric identities
- Solid angle
- Cross product and Seven-dimensional cross product
- Graded algebra
- Exterior derivative
- Differential geometry
- Volume integral
- Measure (mathematics)
- Product integral
- Gilad Lerman and Tyler Whitehouse On d-dimensional d-semimetrics and simplex-type inequalities for high-dimensional sine functions
- Eriksson, F. "The Law of Sines for Tetrahedra and n-Simplices." Geometriae Dedicata volume 7, pages 71–80, 1978.
- Leonhard Euler, "De mensura angulorum solidorum", in Leonhardi Euleri Opera Omnia, volume 26, pages 204–223.