114學測數學A單選第6題詳解

1. 因為 $\overset{\large\rightharpoonup}{u}$、$\overset{\large\rightharpoonup}{v}$、$\overset{\large\rightharpoonup}{w}$ 彼此互相垂直，所以 $$|\overset{\large\rightharpoonup}{u}-\overset{\large\rightharpoonup}{v}|^2=|\overset{\large\rightharpoonup}{u}|^2+|\overset{\large\rightharpoonup}{v}|^2=2^2+(-1)^2+0^2=5,$$ $$|\overset{\large\rightharpoonup}{v}-\overset{\large\rightharpoonup}{w}|^2=|\overset{\large\rightharpoonup}{v}|^2+|\overset{\large\rightharpoonup}{w}|^2=(-1)^2+2^2+3^2=14.$$ 2. 計算內積： $$(\overset{\large\rightharpoonup}{u}-\overset{\large\rightharpoonup}{v})\cdot(\overset{\large\rightharpoonup}{v}-\overset{\large\rightharpoonup}{w})=(2)(-1)+(-1)(2)+(0)(3)=-4.$$ 另一方面，展開得 $$\overset{\large\rightharpoonup}{u}\cdot\overset{\large\rightharpoonup}{v}-\overset{\large\rightharpoonup}{u}\cdot\overset{\large\rightharpoonup}{w}-|\overset{\large\rightharpoonup}{v}|^2+\overset{\large\rightharpoonup}{v}\cdot\overset{\large\rightharpoonup}{w}.$$ 因為三向量互相垂直，所有內積項為 $0$，所以上式 $=-|\overset{\large\rightharpoonup}{v}|^2$，得 $|\overset{\large\rightharpoonup}{v}|^2=4$。 3. 因此 $|\overset{\large\rightharpoonup}{u}|^2=5-4=1$，$|\overset{\large\rightharpoonup}{w}|^2=14-4=10$。即 $|\overset{\large\rightharpoonup}{u}|=1$，$|\overset{\large\rightharpoonup}{v}|=2$，$|\overset{\large\rightharpoonup}{w}|=\sqrt{10}$。 4. 因為三向量互相垂直，平行六面體的體積為三邊長的乘積： $$V=|\overset{\large\rightharpoonup}{u}|\times|\overset{\large\rightharpoonup}{v}|\times|\overset{\large\rightharpoonup}{w}|=1\times2\times\sqrt{10}=2\sqrt{10}.$$ 故選 $(3)$。