110學測數學單選第4題詳解

1. 設 $D=|\det(\overset{\large\rightharpoonup}{a},\overset{\large\rightharpoonup}{b})|$。 2. 因為兩向量 $\overset{\large\rightharpoonup}{u}=2\overset{\large\rightharpoonup}{a}+\overset{\large\rightharpoonup}{b}$ 與 $\overset{\large\rightharpoonup}{v}=\overset{\large\rightharpoonup}{a}+2\overset{\large\rightharpoonup}{b}$ 所張成的三角形面積為 $$\dfrac{1}{2}|\det(\overset{\large\rightharpoonup}{u},\overset{\large\rightharpoonup}{v})|=\dfrac{1}{2}|\det(2\overset{\large\rightharpoonup}{a}+\overset{\large\rightharpoonup}{b},\;\overset{\large\rightharpoonup}{a}+2\overset{\large\rightharpoonup}{b})|$$ 由行列式的線性性質： $$|\det(2\overset{\large\rightharpoonup}{a}+\overset{\large\rightharpoonup}{b},\;\overset{\large\rightharpoonup}{a}+2\overset{\large\rightharpoonup}{b})|=|2\cdot2-1\cdot1|\cdot|\det(\overset{\large\rightharpoonup}{a},\overset{\large\rightharpoonup}{b})|=3D$$ 所以 $\frac{1}{2}\cdot3D=6$，得 $D=4$。 3. 同理，$\overset{\large\rightharpoonup}{u'}=3\overset{\large\rightharpoonup}{a}+\overset{\large\rightharpoonup}{b}$ 與 $\overset{\large\rightharpoonup}{v'}=\overset{\large\rightharpoonup}{a}+3\overset{\large\rightharpoonup}{b}$ 所張成的三角形面積為 $$\dfrac{1}{2}|\det(3\overset{\large\rightharpoonup}{a}+\overset{\large\rightharpoonup}{b},\;\overset{\large\rightharpoonup}{a}+3\overset{\large\rightharpoonup}{b})|=\dfrac{1}{2}|3\cdot3-1\cdot1|D=\dfrac{1}{2}\cdot8\cdot4=16$$ 故選 $(5)$。