{"id":434,"date":"2012-06-20T00:36:03","date_gmt":"2012-06-19T15:36:03","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=434"},"modified":"2021-09-30T08:42:40","modified_gmt":"2021-09-29T23:42:40","slug":"kbr201204","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kbr201204\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2012\uff1a\u7b2c4\u554f"},"content":{"rendered":"
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\u56db\u9762\u4f53 \\(OABC\\) \u306b\u304a\u3044\u3066, \u6b21\u304c\u6e80\u305f\u3055\u308c\u3066\u3044\u308b\u3068\u3059\u308b.\r\n\\[\r\n\\overrightarrow{OA} \\cdot \\overrightarrow{OB} = \\overrightarrow{OB} \\cdot \\overrightarrow{OC} = \\overrightarrow{OC} \\cdot \\overrightarrow{OA}\r\n\\]\r\n\u70b9 \\(A , B , C\\) \u3092\u901a\u308b\u5e73\u9762\u3092 \\(\\alpha\\) \u3068\u3059\u308b.\r\n\u70b9 \\(O\\) \u3092\u901a\u308a\u5e73\u9762 \\(\\alpha\\) \u3068\u76f4\u4ea4\u3059\u308b\u76f4\u7dda\u3068\u5e73\u9762 \\(\\alpha\\) \u3068\u306e\u4ea4\u70b9\u3092 \\(H\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(\\overrightarrow{OA}\\) \u3068 \\(\\overrightarrow{BC}\\) \u306f\u5782\u76f4\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u70b9 \\(H\\) \u306f \\(\\triangle ABC\\) \u306e\u5782\u5fc3\u3067\u3042\u308b\u3053\u3068, \u3059\u306a\u308f\u3061 \\(\\overrightarrow{AH} \\perp \\overrightarrow{BC}\\) , \\(\\overrightarrow{BH} \\perp \\overrightarrow{CA}\\) , \\(\\overrightarrow{CH} \\perp \\overrightarrow{AB}\\) \u3092\u793a\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(\\left| \\overrightarrow{OA} \\right| = \\left| \\overrightarrow{OB} \\right| = \\left| \\overrightarrow{OC} \\right| = 2\\) , \\(\\overrightarrow{OA} \\cdot \\overrightarrow{OB} = \\overrightarrow{OB} \\cdot \\overrightarrow{OC} = \\overrightarrow{OC} \\cdot \\overrightarrow{OA} =1\\) \u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \\(\\triangle ABC\\) \u306e\u5404\u8fba\u306e\u9577\u3055\u304a\u3088\u3073\u7dda\u5206 \\(OH\\) \u306e\u9577\u3055\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    (1)<\/strong><\/p>\r\n

    \\[\\begin{align}\r\n\\overrightarrow{OA} \\cdot \\overrightarrow{BC} & = \\overrightarrow{OA} \\cdot \\left( \\overrightarrow{OC} -\\overrightarrow{OB} \\right) \\\\\r\n& =\\overrightarrow{OA} \\cdot \\overrightarrow{OC} -\\overrightarrow{OA} \\cdot \\overrightarrow{OB} = 0\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\overrightarrow{OA} \\perp \\overrightarrow{BC}\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n

    (1)<\/strong> \u3068\u540c\u69d8\u306b\u3059\u308c\u3070\r\n\\[\r\n\\overrightarrow{OB} \\perp \\overrightarrow{CA} , \\ \\overrightarrow{OC} \\perp \\overrightarrow{AB} \\quad ... [1]\r\n\\]\r\n\u307e\u305f, \u6761\u4ef6\u3088\u308a\r\n\\[\r\n\\overrightarrow{OH} \\perp \\overrightarrow{AB} , \\ \\overrightarrow{OH} \\perp \\overrightarrow{BC} , \\ \\overrightarrow{OH} \\perp \\overrightarrow{CA} \\quad ... [2]\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n\\overrightarrow{AH} \\cdot \\overrightarrow{BC} & = \\left( \\overrightarrow{OH} -\\overrightarrow{OA} \\right) \\cdot \\overrightarrow{BC} \\\\\r\n& = \\overrightarrow{OH} \\cdot \\overrightarrow{BC} -\\overrightarrow{OA} \\cdot \\overrightarrow{BC} = 0 \\\\\r\n\\text{\u2234} \\quad & \\overrightarrow{AH} \\perp \\overrightarrow{BC}\r\n\\end{align}\\]\r\n[1] [2] \u3092\u7528\u3044\u3066\u540c\u69d8\u306b\u3059\u308c\u3070\r\n\\[\r\n\\overrightarrow{BH} \\perp \\overrightarrow{CA} , \\ \\overrightarrow{CH} \\perp \\overrightarrow{AB}\r\n\\]\r\n

    (3)<\/strong><\/p>\r\n

    \\[\\begin{align}\r\n\\left| \\overrightarrow{AB} \\right| & = \\sqrt{\\left| \\overrightarrow{OA} \\right|^2 +\\left| \\overrightarrow{OB} \\right|^2 -2 \\overrightarrow{OA} \\cdot \\overrightarrow{OB}} \\\\\r\n& = \\sqrt{4+4-2 \\cdot 1} = \\sqrt{6}\r\n\\end{align}\\]\r\n\u540c\u69d8\u306b\u3057\u3066\r\n\\[\r\nAB = BC = CA = \\underline{\\sqrt{6}}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(\\triangle ABC\\) \u306f\u6b63\u4e09\u89d2\u5f62\u3067\u3042\u308a, \u5782\u5fc3 \\(H\\) \u306f\u91cd\u5fc3\u3068\u4e00\u81f4\u3059\u308b\u306e\u3067\r\n\\[\r\n\\overrightarrow{OH} = \\dfrac{1}{3} \\left( \\overrightarrow{OA} +\\overrightarrow{OB} +\\overrightarrow{OC} \\right)\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{OH} \\right| & = \\dfrac{1}{3} \\sqrt{3 \\left| \\overrightarrow{OA} \\right|^2 +3 \\cdot 2 \\overrightarrow{OA} \\cdot \\overrightarrow{OB}} \\\\\r\n& =\\dfrac{1}{3} \\sqrt{12+6} = \\underline{\\sqrt{2}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u56db\u9762\u4f53 \\(OABC\\) \u306b\u304a\u3044\u3066, \u6b21\u304c\u6e80\u305f\u3055\u308c\u3066\u3044\u308b\u3068\u3059\u308b. \\[ \\overrightarrow{OA} \\cdot \\overrightarrow{OB} = \\overrightarrow{OB} \\cdot … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[65],"tags":[144,68],"class_list":["post-434","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2012","tag-tsukuba_r","tag-68"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/434","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=434"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/434\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=434"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=434"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=434"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}