こんにちは。今回は2直線のなす角をベクトルを用いて求めてみましょう。
【例題】2直線,
のなす角
を求めよ。ただし,
は鋭角とする。
【解法①:方向ベクトルを用いる場合】先ずは与式をについて解き, グラフを描いて様子を見る。
2つのグラフは
![Rendered by QuickLaTeX.com y=\dfrac12 x+1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-7ac35957bf7a02a63fe58827fdf4e6ea_l3.png)
![Rendered by QuickLaTeX.com y=3x+2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-e5eaed185c37cbd7ab17c46aca66dbe0_l3.png)
![Rendered by QuickLaTeX.com \cos\theta](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-2fb83ed4d91d119004b0ac6342573c3f_l3.png)
![Rendered by QuickLaTeX.com \overrightarrow{\mathstrut a}=(2, 1), \overrightarrow{\mathstrut b}=(1, 3)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-75fd0f9ab13f21f353b89e56b7f4df29_l3.png)
![Rendered by QuickLaTeX.com \overrightarrow{\mathstrut a}\cdot\overrightarrow{\mathstrut b}=2\cdot1+1\cdot3=5](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-19696458476e7ca9f289f89b9951e36b_l3.png)
![Rendered by QuickLaTeX.com |\overrightarrow{\mathstrut a}|=\sqrt{2^2+1^2}=\sqrt5](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-37c4d6dc88eb73768ea9a3baa29006a3_l3.png)
![Rendered by QuickLaTeX.com |\overrightarrow{\mathstrut b}|=\sqrt{1^2+3^2}=\sqrt{10}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-a7df39921fffb61add4d9cd277b0e493_l3.png)
内積の公式より,
![Rendered by QuickLaTeX.com \begin{array}{lll}\cos\theta&=&\dfrac{\overrightarrow{ \mathstrut a}\cdot\overrightarrow{ \mathstrut b}}{| \overrightarrow{ \mathstrut a} | | \overrightarrow{ \mathstrut b} |}\\&=&\dfrac{5}{\sqrt5\cdot\sqrt{10}}\\&=&\dfrac{1}{\sqrt2}\end{array}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4199fff3a1aaaf791a9788fbfd251eac_l3.png)
![Rendered by QuickLaTeX.com 0<\theta<\dfrac{\pi}{2}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-385374a81e3363c75c729eb509446022_l3.png)
より,
![Rendered by QuickLaTeX.com \theta=\dfrac{\pi}{4}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-dc55bd352e8854d30642bf535a72b2ab_l3.png)
【解法②:法線ベクトルを用いる場合】
2直線
![Rendered by QuickLaTeX.com x-2y+2=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-a60f6505674cc6b8423f44e8fd7aa6da_l3.png)
![Rendered by QuickLaTeX.com 3x-y+2=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-8b09089c9f931cb168bb2bd61c4af81e_l3.png)
![Rendered by QuickLaTeX.com \overrightarrow{ \mathstrut a}=(1, -2)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f62c678bd717cdaf542a5e22194eb6ca_l3.png)
![Rendered by QuickLaTeX.com \overrightarrow{ \mathstrut b}=(3, -1)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d07ee624502f9984129f720faf060151_l3.png)
![Rendered by QuickLaTeX.com \overrightarrow{ \mathstrut a}\cdot\overrightarrow{ \mathstrut b}=1\cdot3+(-2)\cdot(-1)=5](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-a4c4c385fa1c759bbe17baba8e158cd0_l3.png)
また,
![Rendered by QuickLaTeX.com |\overrightarrow{ \mathstrut a}|=\sqrt{1^2+(-2)^2}=\sqrt5](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-a11225ce510250bdaef839d4626f364c_l3.png)
![Rendered by QuickLaTeX.com |\overrightarrow{ \mathstrut b}|=\sqrt{3^2+(-1)^2}=\sqrt{10}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d80821a92e7719b0fc30e241b6975ad5_l3.png)
内積の公式より,
![Rendered by QuickLaTeX.com \begin{array}{lll}\cos\theta&=&\dfrac{\overrightarrow{ \mathstrut a}\cdot\overrightarrow{ \mathstrut b}}{| \overrightarrow{ \mathstrut a} | | \overrightarrow{ \mathstrut b} |}\\&=&\dfrac{5}{\sqrt5\cdot\sqrt{10}}\\&=&\dfrac{1}{\sqrt2}\end{array}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4199fff3a1aaaf791a9788fbfd251eac_l3.png)
![Rendered by QuickLaTeX.com 0<\theta<\dfrac{\pi}{2}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-385374a81e3363c75c729eb509446022_l3.png)
より,
![Rendered by QuickLaTeX.com \theta=\dfrac{\pi}{4}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-dc55bd352e8854d30642bf535a72b2ab_l3.png)
こんな感じで求めていきます。
こんにちは。今回は2直線のなす角をベクトルを用いて求めてみましょう。
【例題】2直線,
のなす角
を求めよ。ただし,
は鋭角とする。
【解法①:方向ベクトルを用いる場合】先ずは与式をについて解き, グラフを描いて様子を見る。
こんな感じで求めていきます。