IEVref: 102-03-37 ID: Language: en Status: Standard Term: determinant, n vectors> Synonym1: Synonym2: Synonym3: Symbol: Definition: for an ordered set of n vectors in an n-dimensional space with a given base, scalar attributed to this set by the unique multilinear form taking the value 0 when the vectors are linearly dependent and the value 1 for the base vectorsNote 1 to entry: When the coordinates of the n vectors ${U}_{1}\text{,}{U}_{2}\text{,}\dots ,\text{}{U}_{n}$ are arranged as columns or rows of an $n×n$ matrix, the determinant of the vectors is equal to the determinant of the matrix: $\mathrm{det}\text{\hspace{0.17em}}\left({U}_{1}\text{,}{U}_{2}\text{,}\dots \text{,}{U}_{n}\text{)}=|\begin{array}{cccc}{U}_{11}& {U}_{12}& \cdots & {U}_{1n}\\ {U}_{21}& {U}_{22}& \cdots & {U}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {U}_{n1}& {U}_{n2}& \cdots & {U}_{nn}\end{array}|$ 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. Publication date: 2008-08 Source: Replaces: Internal notes: 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO CO remarks: TC/SC remarks: VT remarks: Domain1: Domain2: Domain3: Domain4: Domain5: