首先利用餘弦定理計算 $\cos A$ 的值： $$\cos A = \dfrac{\overline{AB}^2 + \overline{AC}^2 - \overline{BC}^2}{2 \times \overline{AB} \times \overline{AC}} = \dfrac{5^2 + 6^2 - 8^2}{2 \times 5 \times 6} = \dfrac{25 + 36 - 64}{60} = -\dfrac{3}{60} = -0.05$$ 計算向量 $\overset{\large\rightharpoonup}{AB}$ 與 $\overset{\large\rightharpoonup}{AC}$ 的內積： $$\overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{AC} = |\overset{\large\rightharpoonup}{AB}| |\overset{\large\rightharpoonup}{AC}| \cos A = 5 \times 6 \times (-0.05) = -1.5$$ 接下來計算各選項與 $\overset{\large\rightharpoonup}{AB}$ 的內積值： - $(1)$ $\overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{AC} = -1.5$ - $(2)$ $\overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{CA} = -\overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{AC} = 1.5$ - $(3)$ $\overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{BC} = \overset{\large\rightharpoonup}{AB} \cdot (\overset{\large\rightharpoonup}{AC} - \overset{\large\rightharpoonup}{AB}) = \overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{AC} - |\overset{\large\rightharpoonup}{AB}|^2 = -1.5 - 5^2 = -26.5$ - $(4)$ $\overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{CB} = \overset{\large\rightharpoonup}{AB} \cdot (\overset{\large\rightharpoonup}{AB} - \overset{\large\rightharpoonup}{AC}) = |\overset{\large\rightharpoonup}{AB}|^2 - \overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{AC} = 25 - (-1.5) = 26.5$ - $(5)$ $\overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{AB} = |\overset{\large\rightharpoonup}{AB}|^2 = 25$ 比較所有內積值，以選項 $(4)$ 的內積值 $26.5$ 為最大。故選 $(4)$。