The cross product in three-dimensional Euclidean space is theoretically defined as the unique dual vector associated with a specific linear transformation from $\mathbb{R}^3$ to the scalar field, where this transformation computes the signed volume of a parallelepiped formed by an input variable vector and two fixed vectors. This mechanism relies on the duality principle, which states that any linear map to the number line can be represented equivalently as a dot product with a specific coefficient vector derived from the matrix's cofactors. The resulting dual vector possesses geometric properties such as orthogonality to both source vectors and a magnitude equal to the area of the parallelogram they span, thereby unifying algebraic determinant computations with intrinsic geometric volume measurements within linear algebra theory.
The cross product in three-dimensional Euclidean space is theoretically defined as the unique dual vector associated with a specific linear transformation from $\mathbb{R}^3$ to the scalar field, whe…