<單選題>坐標平面上有兩向量 \(\overset{\rightharpoonup}{u} = (5,10) \)，\(\overset{\rightharpoonup}{v} = (-4,2)\)。請問下列哪一個向量的長度最大？
(1) \(-3 \overset{\rightharpoonup}{u}\)
(2) \(6 \overset{\rightharpoonup}{v}\)
(3) \(-2 \overset{\rightharpoonup}{u} – 5 \overset{\rightharpoonup}{v}\)
(4) \(2 \overset{\rightharpoonup}{u} – 5 \overset{\rightharpoonup}{v}\)
(5) \(\overset{\rightharpoonup}{u} + 7 \overset{\rightharpoonup}{v}\)
答案
(1)。
1. \( -3\vec{u} = (-15, -30) \)
\[
|-3\vec{u}| = \sqrt{(-15)^2 + (-30)^2} = 15\sqrt{5}
\]
2. \( 6\vec{v} = (-24, 12) \)
\[
|6\vec{v}| = \sqrt{(-24)^2 + 12^2} = 12\sqrt{5}
\]
3. \( -2\vec{u} - 5\vec{v} = (10, -30) \)
\[
|-2\vec{u} - 5\vec{v}| = \sqrt{10^2 + (-30)^2} = 10\sqrt{10}
\]
4. \( 2\vec{u} - 5\vec{v} = (30, 10) \)
\[
|2\vec{u} - 5\vec{v}| = \sqrt{30^2 + 10^2} = 10\sqrt{10}
\]
5. \( \vec{u} + 7\vec{v} = (-23, 24) \)
\[
|\vec{u} + 7\vec{v}| = \sqrt{(-23)^2 + 24^2} = \sqrt{1105}
\]
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