113學測數學B選填第15題詳解

1. 由 $\overset{\large\rightharpoonup}{P_1R}=4\overset{\large\rightharpoonup}{P_1Q_1}$ 得 $$\overset{\large\rightharpoonup}{P_1Q_1}=\frac{1}{4}\overset{\large\rightharpoonup}{P_1R}.$$ 由 $\overset{\large\rightharpoonup}{P_2R}=7\overset{\large\rightharpoonup}{P_2Q_2}$ 得 $$\overset{\large\rightharpoonup}{P_2Q_2}=\frac{1}{7}\overset{\large\rightharpoonup}{P_2R}.$$ 2. 計算 $\overset{\large\rightharpoonup}{Q_1Q_2}$： $$\overset{\large\rightharpoonup}{Q_1Q_2}=\overset{\large\rightharpoonup}{Q_1R}+\overset{\large\rightharpoonup}{RQ_2}.$$ 因為 $\overset{\large\rightharpoonup}{Q_1R}=\overset{\large\rightharpoonup}{P_1R}-\overset{\large\rightharpoonup}{P_1Q_1}=4\overset{\large\rightharpoonup}{P_1Q_1}-\overset{\large\rightharpoonup}{P_1Q_1}=3\overset{\large\rightharpoonup}{P_1Q_1}$， 且 $\overset{\large\rightharpoonup}{RQ_2}=-\overset{\large\rightharpoonup}{Q_2R}=-(\overset{\large\rightharpoonup}{P_2R}-\overset{\large\rightharpoonup}{P_2Q_2})=-(7\overset{\large\rightharpoonup}{P_2Q_2}-\overset{\large\rightharpoonup}{P_2Q_2})=-6\overset{\large\rightharpoonup}{P_2Q_2}$。 3. 所以 $$\overset{\large\rightharpoonup}{Q_1Q_2}=3\overset{\large\rightharpoonup}{P_1Q_1}-6\overset{\large\rightharpoonup}{P_2Q_2}.$$ 故填 $3$、$-6$。