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According to Sarrus' rule, the determinant of a 3×3 matrix involves multiplications between matrix elements identified by crossed diagonals. The cross product is also called the vector product or Gibbs' vector product. The name Gibbs' vector product is after Josiah Willard Gibbs, who around 1881 introduced both the dot product and the cross product, using a dot (a · b) and a cross (a × b) to denote them. To emphasize the fact that the result of a dot product is a scalar, while the result of a cross product is avector, Gibbs also introduced the alternative names scalar product and vector product for the two operations. These alternative names are still widely used in the literature. Both the cross notation (a × b) and the name cross product were possibly inspired by the fact that each scalar component of a × b is computed by multiplying non-corresponding components of a and b. Conversely, a dot product a · b involves multiplications between corresponding components of a and b. As explained below, the cross product can be defined as the determinant of a special 3×3 matrix. According to Sarrus' rule, this involves multiplications between matrix elements identified by crossed diagonals.
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