Cube to Parallelopiped in 3D

Continued the topics I left off on yesterday. One concept that clicked for me was that the determinant of a matrix represents the scaling factor of a transformation. If the transformation reverses the orientation of the space, the determinant becomes negative.

In 2D, a unit square is transformed into a parallelogram, and the determinant tells us how the area is scaled.

In 3D, the determinant represents the scaling of volume. A cube transforms into a parallelepiped (a new term I learned today!), and the determinant gives the volume scaling factor.

Another interesting insight is that when the determinant is 0, the transformation collapses the 3D volume into a lower-dimensional object, such as a plane, meaning there is no volume left :)