設兩向量 $\overset{\large\rightharpoonup}{u}, \overset{\large\rightharpoonup}{v}$ 的夾角為 $120^\circ$。則內積 $\overset{\large\rightharpoonup}{u} \cdot \overset{\large\rightharpoonup}{v} = \left\|\overset{\large\rightharpoonup}{u}\right\| \left\|\overset{\large\rightharpoonup}{v}\right\| \cos 120^\circ = -\dfrac{1}{2} \left\|\overset{\large\rightharpoonup}{u}\right\| \left\|\overset{\large\rightharpoonup}{v}\right\|$。 由題意 $\overset{\large\rightharpoonup}{u} \perp (\overset{\large\rightharpoonup}{u} + \overset{\large\rightharpoonup}{v})$，得： $$\overset{\large\rightharpoonup}{u} \cdot (\overset{\large\rightharpoonup}{u} + \overset{\large\rightharpoonup}{v}) = \left\|\overset{\large\rightharpoonup}{u}\right\|^2 + \overset{\large\rightharpoonup}{u} \cdot \overset{\large\rightharpoonup}{v} = 0$$ $$\left\|\overset{\large\rightharpoonup}{u}\right\|^2 - \dfrac{1}{2} \left\|\overset{\large\rightharpoonup}{u}\right\| \left\|\overset{\large\rightharpoonup}{v}\right\| = 0$$ 因為 $\overset{\large\rightharpoonup}{u}$ 為非零向量，同除以 $\left\|\overset{\large\rightharpoonup}{u}\right\|$ 得： $$\left\|\overset{\large\rightharpoonup}{u}\right\| = \dfrac{1}{2} \left\|\overset{\large\rightharpoonup}{v}\right\| \implies \left\|\overset{\large\rightharpoonup}{v}\right\| = 2\left\|\overset{\large\rightharpoonup}{u}\right\|$$ (1) 錯誤。$\overset{\large\rightharpoonup}{v}$ 的長度是 $\overset{\large\rightharpoonup}{u}$ 長度的 $2$ 倍。 (2) 正確。$\overset{\large\rightharpoonup}{v} \cdot (\overset{\large\rightharpoonup}{u} + \overset{\large\rightharpoonup}{v}) = \overset{\large\rightharpoonup}{u} \cdot \overset{\large\rightharpoonup}{v} + \left\|\overset{\large\rightharpoonup}{v}\right\|^2 = -\dfrac{1}{2} \left\|\overset{\large\rightharpoonup}{u}\right\| \left\|\overset{\large\rightharpoonup}{v}\right\| + \left\|\overset{\large\rightharpoonup}{v}\right\|^2 = -\dfrac{1}{4} \left\|\overset{\large\rightharpoonup}{v}\right\|^2 + \left\|\overset{\large\rightharpoonup}{v}\right\|^2 = \dfrac{3}{4} \left\|\overset{\large\rightharpoonup}{v}\right\|^2$。 又 $\left\|\overset{\large\rightharpoonup}{u} + \overset{\large\rightharpoonup}{v}\right\|^2 = \left\|\overset{\large\rightharpoonup}{u}\right\|^2 + 2 \overset{\large\rightharpoonup}{u} \cdot \overset{\large\rightharpoonup}{v} + \left\|\overset{\large\rightharpoonup}{v}\right\|^2 = \dfrac{1}{4} \left\|\overset{\large\rightharpoonup}{v}\right\|^2 - \dfrac{1}{2} \left\|\overset{\large\rightharpoonup}{v}\right\|^2 + \left\|\overset{\large\rightharpoonup}{v}\right\|^2 = \dfrac{3}{4} \left\|\overset{\large\rightharpoonup}{v}\right\|^2 \implies \left\|\overset{\large\rightharpoonup}{u} + \overset{\large\rightharpoonup}{v}\right\| = \dfrac{\sqrt{3}}{2} \left\|\overset{\large\rightharpoonup}{v}\right\|$。 設夾角為 $\theta$，則 $\cos \theta = \dfrac{\overset{\large\rightharpoonup}{v} \cdot (\overset{\large\rightharpoonup}{u}+\overset{\large\rightharpoonup}{v})}{\left\|\overset{\large\rightharpoonup}{v}\right\| \left\|\overset{\large\rightharpoonup}{u}+\overset{\large\rightharpoonup}{v}\right\|} = \dfrac{\frac{3}{4} \left\|\overset{\large\rightharpoonup}{v}\right\|^2}{\left\|\overset{\large\rightharpoonup}{v}\right\| \cdot \frac{\sqrt{3}}{2} \left\|\overset{\large\rightharpoonup}{v}\right\|} = \dfrac{\sqrt{3}}{2} \implies \theta = 30^\circ$。 (3) 正確。$\overset{\large\rightharpoonup}{u} \cdot (\overset{\large\rightharpoonup}{u} - \overset{\large\rightharpoonup}{v}) = \left\|\overset{\large\rightharpoonup}{u}\right\|^2 - \overset{\large\rightharpoonup}{u} \cdot \overset{\large\rightharpoonup}{v} = \left\|\overset{\large\rightharpoonup}{u}\right\|^2 + \dfrac{1}{2} \left\|\overset{\large\rightharpoonup}{u}\right\| \left\|\overset{\large\rightharpoonup}{v}\right\| > 0$，故夾角為銳角。 (4) 錯誤。$\overset{\large\rightharpoonup}{v} \cdot (\overset{\large\rightharpoonup}{u} - \overset{\large\rightharpoonup}{v}) = \overset{\large\rightharpoonup}{u} \cdot \overset{\large\rightharpoonup}{v} - \left\|\overset{\large\rightharpoonup}{v}\right\|^2 = -\dfrac{1}{2} \left\|\overset{\large\rightharpoonup}{u}\right\| \left\|\overset{\large\rightharpoonup}{v}\right\| - \left\|\overset{\large\rightharpoonup}{v}\right\|^2 < 0$，故夾角為鈍角。 (5) 錯誤。$\left\|\overset{\large\rightharpoonup}{u} + \overset{\large\rightharpoonup}{v}\right\|^2 = \dfrac{3}{4} \left\|\overset{\large\rightharpoonup}{v}\right\|^2$，而 $\left\|\overset{\large\rightharpoonup}{u} - \overset{\large\rightharpoonup}{v}\right\|^2 = \left\|\overset{\large\rightharpoonup}{u}\right\|^2 - 2\overset{\large\rightharpoonup}{u}\cdot\overset{\large\rightharpoonup}{v} + \left\|\overset{\large\rightharpoonup}{v}\right\|^2 = \dfrac{1}{4}\left\|\overset{\large\rightharpoonup}{v}\right\|^2 + \left\|\overset{\large\rightharpoonup}{v}\right\|^2 + \left\|\overset{\large\rightharpoonup}{v}\right\|^2 = \dfrac{9}{4}\left\|\overset{\large\rightharpoonup}{v}\right\|^2$。 故 $\left\|\overset{\large\rightharpoonup}{u} + \overset{\large\rightharpoonup}{v}\right\| < \left\|\overset{\large\rightharpoonup}{u} - \overset{\large\rightharpoonup}{v}\right\|$。 故選 $(2)(3)$。