107指考數學乙選填第9題詳解

1. 因為 $D$ 為 $\overline{AB}$ 的中點，所以 $\overset{\large\rightharpoonup}{AD}=\frac{1}{2}\overset{\large\rightharpoonup}{AB}$。 2. 已知 $\overset{\large\rightharpoonup}{AC}\cdot\overset{\large\rightharpoonup}{AD}=15$，代入得 $$\overset{\large\rightharpoonup}{AC}\cdot\frac{1}{2}\overset{\large\rightharpoonup}{AB}=15\implies\overset{\large\rightharpoonup}{AB}\cdot\overset{\large\rightharpoonup}{AC}=30$$ 3. 因為 $\overset{\large\rightharpoonup}{AE}=3\overset{\large\rightharpoonup}{AC}$，所以 $$\overset{\large\rightharpoonup}{AB}\cdot\overset{\large\rightharpoonup}{AE}=\overset{\large\rightharpoonup}{AB}\cdot(3\overset{\large\rightharpoonup}{AC})=3(\overset{\large\rightharpoonup}{AB}\cdot\overset{\large\rightharpoonup}{AC})=3\times 30=90$$ 故填 $90$。