2d:null 2e:["$","title",null,{"children":"三角形的「有向面積」 | lochiwei | Observable"}] 33:[] 31:{"id":"097e0534ade86769","slug":"untitled","title":"三角形的「有向面積」","thumbnail":"5a4ee86067388c47fd8a5dfa895bda0a859dde5dd7f2a073a6e51d0768297e55","access_level":"public","create_time":"$D2021-01-14T08:19:56.793Z","update_time":"$D2021-05-09T13:26:10.408Z","view_time":null,"first_public_time":"$D2021-01-20T08:22:04.907Z","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":297,"publish_version":297,"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":"三角形的「有向面積」","cells":[{"id":0,"value":"md`\n# 三角形的「有向面積」\n---\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":130,"value":"myTOC = my.toc","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":116,"value":"md`---\n在定義「多邊形的有向面積」之前，我們先從「三角形」開始。`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":8,"value":"md`---\n## 🔰 三角形面積（定義）\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":98,"value":"definition = md`\n

${mathCSS.tag.definition()}\n我們利用平面向量的「[外積](https://observablehq.com/d/a80341917e0de070)」來定義三角形的面積:\n

\n${local.equ.area()}\n

因為此值有正負，所以我們稱之為三角形的 ${my.key(\"有向面積\")}。

\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":11,"value":"tex.block`\n\\Delta ABC = \\frac{1}{2}\\; \\overrightarrow{AB} \\times \\overrightarrow{AC}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":26,"value":"mathCSS.tag.figure()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":178,"value":"my.embed.ggb(\"vmgqmvjz\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":189,"value":"mathCSS.tag.calc()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":29,"value":"md`\n如果我們將此定義稍加運算，可得到另一種算法：\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":32,"value":"tex.block`\n\\begin{array}{rcl}\n \\overrightharpoon{AB} \\times \\overrightharpoon{AC} \n　&= & (B-A) \\times (C-A) \\\\\n　&= & B \\times C - B \\times A - A \\times C \\\\\n &= & B \\times C + A \\times B + C \\times A \\\\\n &= & A \\times B + B \\times C + C \\times A\n\\end{array}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":38,"value":"md`---\n## 🔰 三角形面積（算法二）\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":122,"value":"method2 = md`\n

\n${mathCSS.tag.theorem()}\n

\n${local.equ.area2()}\n

\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":274,"value":"md`\n從這個定理可以看出來，${tex`\\Delta ABC`}、${tex`\\Delta BCA`}、${tex`\\Delta CAB`} 面積都是一樣的，但 ${tex`\\Delta ACB`}、${tex`\\Delta BAC`}、${tex`\\Delta CBA`} 面積都跟前面那一組差一個負號。\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":58,"value":"md`---\n## 🔰 三角形面積（算法三）\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":142,"value":"method3 = md`\n

\n${mathCSS.tag.theorem()}\n若 ${tex`\\color{ff0000}{X}`} 為 ${tex`\\overleftrightarrow{BC}`} 直線上的任一點，則：\n

\n${local.equ.area3()}\n

\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":280,"value":"md`\n這個公式是一個比較有「彈性」的公式，其中的 ${tex`\\color{ff0000}{X}`} 是直線 ${tex`\\overleftrightarrow{BC}`} 上的任一點，所以可以「${my.marker(\"在線上任意調整\")}」，也不會影響面積值。\n\n當我們有兩個「${my.marker(\"同底\")}」的三角形時，我們可以分別調整這兩個三角形的 ${tex`\\overrightharpoon{A{\\color{red}X}}`} 向量，讓它們平行，這時就可以將兩個三角形的「${my.marker(\"面積比\")}」轉換成兩個向量的「${my.marker(\"向量比\")}」。\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":193,"value":"mathCSS.tag.figure()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":173,"value":"my.embed.ggb(\"prpp6kqn\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":78,"value":"mathCSS.tag.proof()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":196,"value":"md`\n因為 ${tex`\\color{red}{X}`} 在 ${tex`\\overleftrightarrow{AB}`} 直線上，所以：\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":202,"value":"tex.block`\n\\begin{aligned}\n \\overrightharpoon{A\\color{red}{X}} \n &= \\overrightharpoon{AB} + \\overrightharpoon{B\\color{red}{X}} \\\\ \n &= \\overrightharpoon{AB} + {\\color{blue}{t}} \\cdot \\overrightharpoon{BC} \\\\\n &= \\overrightharpoon{AB} + {\\color{blue}{t}} \\cdot (\\overrightharpoon{AC}-\\overrightharpoon{AB}) \\\\\n &= (1-{\\color{blue}{t}}) \\cdot \\overrightharpoon{AB} + {\\color{blue}{t}} \\cdot \\overrightharpoon{AC}\n\\end{aligned}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":216,"value":"md`另外：`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":219,"value":"tex.block`\n\\overrightharpoon{BC} = \\overrightharpoon{AC} - \\overrightharpoon{AB}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":226,"value":"md`因此：`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":229,"value":"tex.block`\n\\begin{aligned}\n \\overrightharpoon{A\\color{red}{X}} \\times \\overrightharpoon{BC}\n &= \\left(\n (1-{\\color{blue}{t}}) \\cdot \\overrightharpoon{AB} + {\\color{blue}{t}} \\cdot \\overrightharpoon{AC} \n \\right) \\times \\left( \\overrightharpoon{AC} - \\overrightharpoon{AB} \\right) \\\\\n &= (1-{\\color{blue}{t}}) \\cdot \\overrightharpoon{AB} \\times \\overrightharpoon{AC}\n - {\\color{blue}{t}} \\cdot \\overrightharpoon{AC} \\times \\overrightharpoon{AB} \\\\\n &= \\overrightharpoon{AB} \\times \\overrightharpoon{AC}\n\\end{aligned}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":262,"value":"md`\n根據上述「${my.link.inner(\"definition\", \"三角形有向面積\")}」的定義：\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":267,"value":"tex.block`\n\\Delta ABC \n = \\frac{1}{2}\\; \\overrightarrow{AB} \\times \\overrightarrow{AC} \n = \\frac{1}{2}\\; \\overrightharpoon{A\\color{red}{X}} \\times \\overrightharpoon{BC} \n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":295,"value":"my.link.toc()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":151,"value":"md`---\n## 附錄`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":81,"value":"md`\n### imports\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":84,"value":"import {mathCSS} from \"@lochiwei/mathcss2\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":87,"value":"mathCSS.style","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":90,"value":"import {my} from \"@lochiwei/my\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":94,"value":"my.style","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":100,"value":"md`\n### vars\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":103,"value":"local = ({\n equ: {\n area: () => tex.block`\\Delta ABC = \\frac{1}{2}\\; \\overrightarrow{AB} \\times \\overrightarrow{AC}`,\n area2: () => tex.block`\\Delta ABC = \\frac{1}{2}\\; (A \\times B + B \\times C + C \\times A)`,\n area3: () => tex.block`\\Delta ABC = \\frac{1}{2}\\; \\overrightharpoon{A\\color{ff0000}{X}} \\times \\overrightharpoon{BC}`\n }\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":169,"value":"my.link.toc()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W33","stdlib":"1","titling":"auto"}} 32:"$31:body"