106指考數學乙選填第8題詳解

1. 因為 $\overset{\large\rightharpoonup}{u}\perp\overset{\large\rightharpoonup}{v}$，所以 $\overset{\large\rightharpoonup}{u}\cdot\overset{\large\rightharpoonup}{v}=0$。 2. 已知 $\overset{\large\rightharpoonup}{u}-\overset{\large\rightharpoonup}{v}=(4,-7)$，取長度的平方： $$|\overset{\large\rightharpoonup}{u}-\overset{\large\rightharpoonup}{v}|^2=4^2+(-7)^2=16+49=65$$ 3. 展開 $|\overset{\large\rightharpoonup}{u}-\overset{\large\rightharpoonup}{v}|^2$： $$|\overset{\large\rightharpoonup}{u}-\overset{\large\rightharpoonup}{v}|^2=|\overset{\large\rightharpoonup}{u}|^2-2\overset{\large\rightharpoonup}{u}\cdot\overset{\large\rightharpoonup}{v}+|\overset{\large\rightharpoonup}{v}|^2=65$$ 4. 代入 $|\overset{\large\rightharpoonup}{u}|=6$ 及 $\overset{\large\rightharpoonup}{u}\cdot\overset{\large\rightharpoonup}{v}=0$： $$36-0+|\overset{\large\rightharpoonup}{v}|^2=65\implies|\overset{\large\rightharpoonup}{v}|^2=29$$ 5. 因為向量長度為正，所以 $|\overset{\large\rightharpoonup}{v}|=\sqrt{29}$。 故填 $\sqrt{29}$。