The interpretations of 3d volumes for left: a parallelepiped (Ω in polar sine definition) and right: a cuboid (Π in definition). The interpretation is similar in higher dimensions.
Let v1, ..., vn, for n ≥ 2, be non-zero Euclidean vectors in n-dimensional space (ℝn) that are directed from a vertex of a parallelotope, forming the edges of the parallelotope. The polar sine of the vertex angle is:
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):
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 case n = 2, the polar sine is the ordinary sine of the angle between the two vectors.
A non-negative version of the polar sine exists, which works in any m-dimensional space for m ≥ n. In this case, the numerator in the definition is given as
where the superscript T indicates matrix transposition. In the case that m=n, the value of Ω for this non-negative definition of the polar sine is the absolute value of the Ω from the signed version of the polar sine given previously.
If the dimension of the space is more than n then the polar sine is non-negative and is unchanged whenever two of the vectors vj and vk are interchanged. Otherwise, it changes sign whenever two vectors are interchanged - due to the antisymmetry of row-exchanging in the determinant: