Note 1 to entry: When the coordinates of the n vectors $U_1\text{,}{U}_2\text{,}\dots ,U_n$ are arranged as columns or rows of an $n\times n$ matrix, the determinant of the vectors is equal to the determinant of the matrix:

$\det (U_1\text{,}{U}_2\text{,}\dots \text{,}{U}_n\text{)}=\left|\begin{array}{cccc}{U}_{11}& U_{12}& \cdots & U_{1n}\\ U_{21}& U_{22}& \cdots & U_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ U_{n1}& U_{n2}& \cdots & U_{nn}\end{array}\right|$

Note 2 to entry: According to the sign of the determinant, the set of vectors and the given base have the same orientation or opposite orientations.

Note 3 to entry: For the three-dimensional Euclidean space, the determinant of three vectors is the scalar triple product of the vectors.