Volume & Determinant

Replacing v₁ by v₁ + λv₂ slides the solid sideways, parallel to the base spanned by v₂ and v₃. The base does not move and the height above it does not change, so every slice taken parallel to the base is the same shape as before, merely shifted. Equal cross-sections at every height force equal volume: this is Cavalieri's principle, and a shear is the cleanest example of it.

You can already see it above: drag λ and the box leans further and further over, while the number does not move.