106指考數學甲多選第5題詳解

1. 設 $\overset{\large\rightharpoonup}{u}$ 與 $\overset{\large\rightharpoonup}{v}$ 的夾角為 $120^\circ$，則內積 $$\overset{\large\rightharpoonup}{u}\cdot\overset{\large\rightharpoonup}{v}=|\overset{\large\rightharpoonup}{u}||\overset{\large\rightharpoonup}{v}|\cos 120^\circ=-\frac{1}{2}|\overset{\large\rightharpoonup}{u}||\overset{\large\rightharpoonup}{v}|$$ 2. 因為 $\overset{\large\rightharpoonup}{u}\perp(\overset{\large\rightharpoonup}{u}+\overset{\large\rightharpoonup}{v})$，所以 $$\overset{\large\rightharpoonup}{u}\cdot(\overset{\large\rightharpoonup}{u}+\overset{\large\rightharpoonup}{v})=|\overset{\large\rightharpoonup}{u}|^2+\overset{\large\rightharpoonup}{u}\cdot\overset{\large\rightharpoonup}{v}=|\overset{\large\rightharpoonup}{u}|^2-\frac{1}{2}|\overset{\large\rightharpoonup}{u}||\overset{\large\rightharpoonup}{v}|=0$$ 因為 $\overset{\large\rightharpoonup}{u}$ 為非零向量，除以 $|\overset{\large\rightharpoonup}{u}|$ 得 $|\overset{\large\rightharpoonup}{u}|=\frac{1}{2}|\overset{\large\rightharpoonup}{v}|$，即 $|\overset{\large\rightharpoonup}{v}|=2|\overset{\large\rightharpoonup}{u}|$。 選項 $(1)$：$\overset{\large\rightharpoonup}{u}$ 的長度是 $\overset{\large\rightharpoonup}{v}$ 的 $\frac{1}{2}$ 倍，不是 $2$ 倍。錯誤。 選項 $(2)$：計算 $\overset{\large\rightharpoonup}{v}\cdot(\overset{\large\rightharpoonup}{u}+\overset{\large\rightharpoonup}{v})$： $$\overset{\large\rightharpoonup}{v}\cdot(\overset{\large\rightharpoonup}{u}+\overset{\large\rightharpoonup}{v})=\overset{\large\rightharpoonup}{u}\cdot\overset{\large\rightharpoonup}{v}+|\overset{\large\rightharpoonup}{v}|^2=-\frac{1}{4}|\overset{\large\rightharpoonup}{v}|^2+|\overset{\large\rightharpoonup}{v}|^2=\frac{3}{4}|\overset{\large\rightharpoonup}{v}|^2$$ 又 $$|\overset{\large\rightharpoonup}{u}+\overset{\large\rightharpoonup}{v}|^2=|\overset{\large\rightharpoonup}{u}|^2+2\overset{\large\rightharpoonup}{u}\cdot\overset{\large\rightharpoonup}{v}+|\overset{\large\rightharpoonup}{v}|^2=\frac{1}{4}|\overset{\large\rightharpoonup}{v}|^2-\frac{1}{2}|\overset{\large\rightharpoonup}{v}|^2+|\overset{\large\rightharpoonup}{v}|^2=\frac{3}{4}|\overset{\large\rightharpoonup}{v}|^2$$ 所以 $|\overset{\large\rightharpoonup}{u}+\overset{\large\rightharpoonup}{v}|=\frac{\sqrt{3}}{2}|\overset{\large\rightharpoonup}{v}|$。 設 $\overset{\large\rightharpoonup}{v}$ 與 $\overset{\large\rightharpoonup}{u}+\overset{\large\rightharpoonup}{v}$ 夾角為 $\theta$，則 $$\cos\theta=\frac{\overset{\large\rightharpoonup}{v}\cdot(\overset{\large\rightharpoonup}{u}+\overset{\large\rightharpoonup}{v})}{|\overset{\large\rightharpoonup}{v}||\overset{\large\rightharpoonup}{u}+\overset{\large\rightharpoonup}{v}|}=\frac{\frac{3}{4}|\overset{\large\rightharpoonup}{v}|^2}{|\overset{\large\rightharpoonup}{v}|\cdot\frac{\sqrt{3}}{2}|\overset{\large\rightharpoonup}{v}|}=\frac{\sqrt{3}}{2}$$ 因此 $\theta=30^\circ$。正確。 選項 $(3)$：$\overset{\large\rightharpoonup}{u}\cdot(\overset{\large\rightharpoonup}{u}-\overset{\large\rightharpoonup}{v})=|\overset{\large\rightharpoonup}{u}|^2-\overset{\large\rightharpoonup}{u}\cdot\overset{\large\rightharpoonup}{v}=|\overset{\large\rightharpoonup}{u}|^2+\frac{1}{2}|\overset{\large\rightharpoonup}{u}||\overset{\large\rightharpoonup}{v}|>0$。因為內積為正，所以夾角為銳角。正確。 選項 $(4)$：$\overset{\large\rightharpoonup}{v}\cdot(\overset{\large\rightharpoonup}{u}-\overset{\large\rightharpoonup}{v})=\overset{\large\rightharpoonup}{u}\cdot\overset{\large\rightharpoonup}{v}-|\overset{\large\rightharpoonup}{v}|^2=-\frac{1}{2}|\overset{\large\rightharpoonup}{u}||\overset{\large\rightharpoonup}{v}|-|\overset{\large\rightharpoonup}{v}|^2<0$。因為內積為負，所以夾角為鈍角。錯誤。 選項 $(5)$：計算兩者長度平方： $$|\overset{\large\rightharpoonup}{u}+\overset{\large\rightharpoonup}{v}|^2=\frac{3}{4}|\overset{\large\rightharpoonup}{v}|^2$$ $$|\overset{\large\rightharpoonup}{u}-\overset{\large\rightharpoonup}{v}|^2=|\overset{\large\rightharpoonup}{u}|^2-2\overset{\large\rightharpoonup}{u}\cdot\overset{\large\rightharpoonup}{v}+|\overset{\large\rightharpoonup}{v}|^2=\frac{1}{4}|\overset{\large\rightharpoonup}{v}|^2+|\overset{\large\rightharpoonup}{v}|^2+|\overset{\large\rightharpoonup}{v}|^2=\frac{7}{4}|\overset{\large\rightharpoonup}{v}|^2$$ 因為 $\frac{3}{4}<\frac{7}{4}$，所以 $|\overset{\large\rightharpoonup}{u}+\overset{\large\rightharpoonup}{v}|<|\overset{\large\rightharpoonup}{u}-\overset{\large\rightharpoonup}{v}|$。錯誤。 故選 $(2)(3)$。