## 2016年7月30日土曜日

### 数学 – 図形と代数の交錯する世界 - 平面上のベクトル – (内積を成分で表すこと, 射影、成分)

• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Edge/Internet Explorer, Google Chrome...)用JavaScript Library: MathJax

1. $\begin{array}{l}\left(\stackrel{\to }{a}-{t}_{0}\stackrel{\to }{b|}\right)·\stackrel{\to }{b}=0\\ \left(3-2{t}_{0},4-{t}_{0}\right)·\left(2,1\right)=0\\ 6-4{t}_{0}+4-{t}_{0}=0\\ {t}_{0}=2\\ {|\stackrel{\to }{a}-2\stackrel{\to }{b}|}^{2}\\ =\left(\stackrel{\to }{a}-2\stackrel{\to }{b}\right)·\left(\stackrel{\to }{a}-2\stackrel{\to }{b}\right)\\ ={|\stackrel{\to }{a}|}^{2}-4\stackrel{\to }{a}·\stackrel{\to }{b}+4{|\stackrel{\to }{b}|}^{2}\\ =25-4·10+4·5\\ =25-40+20\\ =5\\ |\stackrel{\to }{a}-2\stackrel{\to }{b}|=\sqrt{5}\end{array}$

2. $\begin{array}{l}\left(\stackrel{\to }{a}-{t}_{0}\stackrel{\to }{b|}\right)·\stackrel{\to }{b}=0\\ \left(1+{t}_{0},-3-{t}_{0}\right)·\left(-1,1\right)=0\\ -1-{t}_{0}-3-{t}_{0}=0\\ {t}_{0}=-2\\ {|\stackrel{\to }{a}+2\stackrel{\to }{b}|}^{2}\\ =\left(\stackrel{\to }{a}+2\stackrel{\to }{b}\right)·\left(\stackrel{\to }{a}+2\stackrel{\to }{b}\right)\\ ={|\stackrel{\to }{a}|}^{2}+4\stackrel{\to }{a}·\stackrel{\to }{b}+4{|\stackrel{\to }{b}|}^{2}\\ =10-16+8\\ =2\\ |\stackrel{\to }{a}+2\stackrel{\to }{b}|=\sqrt{2}\end{array}$

3. $\begin{array}{l}\left(\stackrel{\to }{a}-{t}_{0}\stackrel{\to }{b|}\right)·\stackrel{\to }{b}=0\\ \stackrel{\to }{a}·\stackrel{\to }{b}-{t}_{0}\stackrel{\to }{b}·\stackrel{\to }{b}=0\\ 4-{t}_{0}{|\stackrel{\to }{b}|}^{2}=0\\ {t}_{0}=1\\ {||\stackrel{\to }{a}-\stackrel{\to }{b}|}^{2}\\ =\left(\stackrel{\to }{a}-\stackrel{\to }{b}\right)·\left(\stackrel{\to }{a}-\stackrel{\to }{b}\right)\\ =\stackrel{\to }{a}·\stackrel{\to }{a}-2\stackrel{\to }{a}·\stackrel{\to }{b}+\stackrel{\to }{b}·\stackrel{\to }{b}\\ ={|\stackrel{\to }{a}|}^{2}-8+{|\stackrel{\to }{b}|}^{2}\\ =5\\ ||\stackrel{\to }{a}-\stackrel{\to }{b}|=\sqrt{5}\end{array}$