A parallelotope is the region spanned by vectors. In two dimensions it is a parallelogram; in three dimensions, a parallelepiped; in higher dimensions, the same idea measures generalized volume. section 9.5.1 section 9.5.2
Vector products expose this geometry: a cross product length gives the volume of an (N−1)-dimensional parallelotope, while a triple product gives signed N-dimensional volume. section 9.5.1
Useful two-dimensional interpretations include:
- The box product of two vectors gives the signed area of their parallelogram. section 9.5.2
- A 2D cross product can place that signed area in the third coordinate, linking it to the 2×2 determinant. section 9.5.2
- For the vector (2, 3), the 2D cross product is (3, −2), using the basis vectors i and j. section 9.5.2
Use the geometric figure to connect spanning vectors with the parallelogram, parallelepiped, and higher-dimensional volume interpretations. section 9.5.2
