2d:null 2e:["$","title",null,{"children":"兩向量的「向量比」 | lochiwei | Observable"}] 33:[] 31:{"id":"1b8f546bb881c946","slug":"vector-ratio","title":"兩向量的「向量比」","thumbnail":"d9ce3f4090c70554789bd4911f5e754cf719a6afb0fba0a7095aac9879a19804","access_level":"public","create_time":"$D2021-01-20T08:30:41.106Z","update_time":"$D2021-01-20T15:18:32.812Z","view_time":null,"first_public_time":"$D2021-01-20T14:24:50.795Z","has_unpublished_edits":null,"fork":false,"stars":1,"starred":false,"owner":{"type":"team","id":"51d0aa8921c02d01","login":"lochiwei","name":"lochiwei","avatar_url":"https://avatars-next.observableusercontent.com/ab27db86392e58dc187a21fef0d6cb4c5387914b82b7134a9da0ede3cc77ad7a","tier":"starter_2024"},"version":120,"publish_version":120,"role":null,"authored":false,"authors":[{"type":"individual","id":"917c8eb3367f9e91","login":"lochiwei","name":"lochiwei","avatar_url":"https://avatars-next.observableusercontent.com/ab27db86392e58dc187a21fef0d6cb4c5387914b82b7134a9da0ede3cc77ad7a","tier":null}],"members":[{"type":"individual","id":"917c8eb3367f9e91","login":"lochiwei","name":"lochiwei","avatar_url":"https://avatars-next.observableusercontent.com/ab27db86392e58dc187a21fef0d6cb4c5387914b82b7134a9da0ede3cc77ad7a","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":{"id":"5e8aa5577f29cf06","slug":"geometry-new-road","title":"📗 幾何心路","owner":{"id":"51d0aa8921c02d01","login":"lochiwei","name":"lochiwei"}},"type":"notebook","body":{"title":"兩向量的「向量比」","cells":[{"id":0,"value":"md`# 兩向量的「向量比」`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5,"value":"md`\n若 ${tex`\\vec{u}\\neq\\vec{0}`}，當 ${tex`\\vec{u}`}、${tex`\\vec{v}`} 兩向量平行時，我們可以寫成 ${tex`\\vec{v} = {\\color{blue}t} \\; \\vec{u}`}，其中 ${tex`{\\color{blue}t} \\in \\R`}。\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":45,"value":"calc1 = mathCSS.tag.calc()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":48,"value":"md`如果我們對這個方程式稍加計算，可得：`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":51,"value":"tex.block`\n\\begin{aligned}\n {\\color{red}\\vec{v}} \\cdot \\vec{u} &= {\\color{blue}t}\\; ( \\vec{u} \\cdot \\vec{u} )\\\\\n {\\color{blue}t} &= \\frac{{\\color{red}\\vec{v}} \\cdot \\vec{u}}{\\vec{u} \\cdot \\vec{u}}\n\\end{aligned}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":62,"value":"md`\n我們可以利用這個算式來定義兩個向量的「向量比」\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":65,"value":"html`\n

${mathCSS.tag.definition()} 兩向量的「向量比」

\n${env.tex.defVectorRatio()}\n

其中 ${tex`\\vec{u}\\neq\\vec{0}`}。

\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":108,"value":"md`\n這個「比值」基本上是用來描述「 ${tex`\\vec{\\color{red}{v}}`} 向量是 ${tex`\\vec{u}`} 向量的幾倍長」。\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":104,"value":"my.tag.caution()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":82,"value":"md`\n在上面的定義中，${tex`\\vec{u}`}、${tex`\\vec{v}`} 兩向量並${my.marker(\"不需要有平行關係\")}，而且也不限於平面向量，空間向量也適用。\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":91,"value":"thm1 = html`\n

${mathCSS.tag.theorem()} 若 ${tex`\\vec{v} = {\\color{blue}t} \\; \\vec{u}`}，其中 ${tex`\\vec{u}\\neq\\vec{0}`}，則：

\n${tex.block`\\frac{{\\color{red}\\vec{v}}}{\\vec{u}}={\\color{blue}t}`}\n

\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":113,"value":"md`\n${mathCSS.tag.proof()} 上方開頭的「${my.link.inner(\"calc1\", \"計算\")}」部分就是這個定理的證明。\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":26,"value":"md`---\n## 附錄`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":30,"value":"md`### imports`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":33,"value":"import {my} from \"@lochiwei/my\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":36,"value":"my.style","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":39,"value":"import {mathCSS} from \"@lochiwei/mathcss2\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":42,"value":"mathCSS.style","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":69,"value":"md`### vars`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":72,"value":"env = ({\n \n // KaTex\n tex: {\n defVectorRatio: () => tex.block`\n \\frac{{\\color{red}\\vec{v}}}{\\vec{u}} \\overset{\\Delta}{=} \n \\frac{{\\color{red}\\vec{v}} \\cdot \\vec{u}}{\\vec{u} \\cdot \\vec{u}}\n `\n }\n \n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W33","stdlib":"1","titling":"auto"}} 32:"$31:body"