Determinant in 2D

encodes the area under the parallelogram of with edge vectors and

Volume and Triple Product in 3D

encodes (signed) volume of parallelepiped with edge vectors :triple_product.png

The above relationship is known as a “triple product formula”.

Note we cannot reverse order of cross product, otherwise we get the wrong sign.

Determinant of a Linear Map

If a matrix encodes a linear map , the determinant measures the change in volume: determinant_volume_change.png