因 $\overset{\large\rightharpoonup}{u},\overset{\large\rightharpoonup}{v},\overset{\large\rightharpoonup}{w}$ 互相垂直，$|\overset{\large\rightharpoonup}{u}-\overset{\large\rightharpoonup}{v}|^2=|\overset{\large\rightharpoonup}{u}|^2+|\overset{\large\rightharpoonup}{v}|^2=5$，$|\overset{\large\rightharpoonup}{v}-\overset{\large\rightharpoonup}{w}|^2=|\overset{\large\rightharpoonup}{v}|^2+|\overset{\large\rightharpoonup}{w}|^2=14$。又 $(\overset{\large\rightharpoonup}{u}-\overset{\large\rightharpoonup}{v})\cdot(\overset{\large\rightharpoonup}{v}-\overset{\large\rightharpoonup}{w})=-4=-|\overset{\large\rightharpoonup}{v}|^2$，故 $|\overset{\large\rightharpoonup}{v}|^2=4$，$|\overset{\large\rightharpoonup}{u}|^2=1$，$|\overset{\large\rightharpoonup}{w}|^2=10$。體積為 $|\overset{\large\rightharpoonup}{u}||\overset{\large\rightharpoonup}{v}||\overset{\large\rightharpoonup}{w}|=2\sqrt{10}$，選 $(3)$。