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.[2]
The parallelotope is like a "squashed hyperrectangle", so it has less hypervolume than the hyperrectangle, meaning (see image for the 3d case):
and since this ratio can be negative, psin is always bounded between −1 and +1 by the inequalities:
as for the ordinary sine, with either bound only being reached in case all vectors are mutually orthogonal.
In the case n = 2, the polar sine is the ordinary sine of the angle between the two vectors.
In higher dimensions
A non-negative version of the polar sine which works in any m-dimensional space (m ≥ n) can be defined using the Gram determinant. The numerator is given as
where the superscript T indicates matrix transposition. In the case m = n this is equivalent to the absolute value of the definition given previously.
Properties
Interchange of vectors
The polar sine changes sign whenever two vectors are interchanged, due to the antisymmetry of row-exchanging in the determinant; however, its absolute value will remain unchanged.
Invariance under scalar multiplication of vectors
The polar sine does not change if all of the vectors v1, ..., vn are scalar-multiplied by positive constants ci, due to factorization
If an odd number of these constants are instead negative, then the sign of the polar sine will change; however, its absolute value will remain unchanged.
Vanishes with linear dependencies
If the vectors are not linearly independent, the polar sine will be zero. This will always be so in the degenerate case that the number of dimensions m is strictly less than the number of vectors n.
Relationship to pairwise cosines
The cosine of the angle between two non-zero vectors is given by
using the dot product. Comparison of this expression to the definition of the absolute value of the polar sine as given above gives:
^Lerman, Gilad; Whitehouse, J. Tyler (2009). "On d-dimensional d-semimetrics and simplex-type inequalities for high-dimensional sine functions". Journal of Approximation Theory. 156: 52–81. arXiv:0805.1430. doi:10.1016/j.jat.2008.03.005. S2CID12794652.