給定兩向量 $\overset{\large\rightharpoonup}{u} = (5,10)$ 與 $\overset{\large\rightharpoonup}{v} = (-4, 2)$。 首先計算此二向量的長度平方： $$\left\|\overset{\large\rightharpoonup}{u}\right\|^2 = 5^2 + 10^2 = 25 + 100 = 125$$ $$\left\|\overset{\large\rightharpoonup}{v}\right\|^2 = (-4)^2 + 2^2 = 16 + 4 = 20$$ 接著，計算這兩個向量的內積： $$\overset{\large\rightharpoonup}{u} \cdot \overset{\large\rightharpoonup}{v} = 5 \times (-4) + 10 \times 2 = -20 + 20 = 0$$ 因為內積為 $0$，所以向量 $\overset{\large\rightharpoonup}{u}$ 與 $\overset{\large\rightharpoonup}{v}$ 互相垂直。 由於 $\overset{\large\rightharpoonup}{u} \perp \overset{\large\rightharpoonup}{v}$，對於任何實數 $x, y$，向量 $x\overset{\large\rightharpoonup}{u} + y\overset{\large\rightharpoonup}{v}$ 的長度平方公式為： $$\left\|x\overset{\large\rightharpoonup}{u} + y\overset{\large\rightharpoonup}{v}\right\|^2 = x^2\left\|\overset{\large\rightharpoonup}{u}\right\|^2 + y^2\left\|\overset{\large\rightharpoonup}{v}\right\|^2$$ 我們分別計算各選項中向量的長度平方： 1. 對於選項 $(1)$：$-3\overset{\large\rightharpoonup}{u}$ $$\left\|-3\overset{\large\rightharpoonup}{u}\right\|^2 = 9\left\|\overset{\large\rightharpoonup}{u}\right\|^2 = 9 \times 125 = 1125$$ 2. 對於選項 $(2)$：$6\overset{\large\rightharpoonup}{v}$ $$\left\|6\overset{\large\rightharpoonup}{v}\right\|^2 = 36\left\|\overset{\large\rightharpoonup}{v}\right\|^2 = 36 \times 20 = 720$$ 3. 對於選項 $(3)$：$-2\overset{\large\rightharpoonup}{u} - 5\overset{\large\rightharpoonup}{v}$ $$\left\|-2\overset{\large\rightharpoonup}{u} - 5\overset{\large\rightharpoonup}{v}\right\|^2 = 4\left\|\overset{\large\rightharpoonup}{u}\right\|^2 + 25\left\|\overset{\large\rightharpoonup}{v}\right\|^2 = 4 \times 125 + 25 \times 20 = 500 + 500 = 1000$$ 4. 對於選項 $(4)$：$2\overset{\large\rightharpoonup}{u} - 5\overset{\large\rightharpoonup}{v}$ $$\left\|2\overset{\large\rightharpoonup}{u} - 5\overset{\large\rightharpoonup}{v}\right\|^2 = 4\left\|\overset{\large\rightharpoonup}{u}\right\|^2 + 25\left\|\overset{\large\rightharpoonup}{v}\right\|^2 = 4 \times 125 + 25 \times 20 = 500 + 500 = 1000$$ 5. 對於選項 $(5)$：$\overset{\large\rightharpoonup}{u} + 7\overset{\large\rightharpoonup}{v}$ $$\left\|\overset{\large\rightharpoonup}{u} + 7\overset{\large\rightharpoonup}{v}\right\|^2 = 1\left\|\overset{\large\rightharpoonup}{u}\right\|^2 + 49\left\|\overset{\large\rightharpoonup}{v}\right\|^2 = 125 + 49 \times 20 = 125 + 980 = 1105$$ 比較所有選項的長度平方值： $$1125 > 1105 > 1000 = 1000 > 720$$ 故長度平方最大的是 $1125$，即向量 $-3\overset{\large\rightharpoonup}{u}$ 的長度最大。 答案選 $(1)$。