105指考數學乙單選第3題詳解

1. 先計算 $\overset{\large\rightharpoonup}{u}=(5,10)$ 與 $\overset{\large\rightharpoonup}{v}=(-4,2)$ 的基本量： $$|\overset{\large\rightharpoonup}{u}|^2 = 5^2+10^2 = 25+100 = 125$$ $$|\overset{\large\rightharpoonup}{v}|^2 = (-4)^2+2^2 = 16+4 = 20$$ 2. 因為 $\overset{\large\rightharpoonup}{u}\cdot\overset{\large\rightharpoonup}{v} = 5\cdot(-4)+10\cdot 2 = -20+20 = 0$，所以 $\overset{\large\rightharpoonup}{u}\perp\overset{\large\rightharpoonup}{v}$。因此對於任意實數 $p,q$，有 $|p\overset{\large\rightharpoonup}{u}+q\overset{\large\rightharpoonup}{v}|^2 = p^2|\overset{\large\rightharpoonup}{u}|^2+q^2|\overset{\large\rightharpoonup}{v}|^2$。 3. 計算各選項向量的長度平方： 選項 $(1)$：$|-3\overset{\large\rightharpoonup}{u}|^2 = 9\cdot 125 = 1125$ 選項 $(2)$：$|6\overset{\large\rightharpoonup}{v}|^2 = 36\cdot 20 = 720$ 選項 $(3)$：$|-2\overset{\large\rightharpoonup}{u}-5\overset{\large\rightharpoonup}{v}|^2 = 4\cdot 125 + 25\cdot 20 = 500+500 = 1000$ 選項 $(4)$：$|2\overset{\large\rightharpoonup}{u}-5\overset{\large\rightharpoonup}{v}|^2 = 4\cdot 125 + 25\cdot 20 = 1000$ 選項 $(5)$：$|\overset{\large\rightharpoonup}{u}+7\overset{\large\rightharpoonup}{v}|^2 = 1\cdot 125 + 49\cdot 20 = 125+980 = 1105$ 4. 因為 $1125 > 1105 > 1000 = 1000 > 720$，所以選項 $(1)$ 的向量 $-3\overset{\large\rightharpoonup}{u}$ 長度最大。故選 $(1)$。