Should the parallelepiped have a parallelogram as the base?
I think since each face of the parallelepiped is spanned by two vectors, each face should be a parallelogram.
Also, there is a typo on the slide: parallelpiped should be parallelepiped.
What does a determinant represent at higher dimensions?
I believe the determinant represents the same thing (volume) at any dimension. Of course it's much harder to visualize at higher dimensions, thus the 3D example.
@siminl Yes; the fact that in this picture they don't look like parallelograms is purely a result of perspective distortion (which, in fact, we will learn about in our class! :-)). But these diagrams were actually generated by creating a 3D model with parallelograms on each side.
@ananyas Thanks for catching the typo; fixed for next year! ;-)
@ericchan Yes, @adam is totally right: the determinant gives you the volume in any dimension.