{"id":865,"date":"2013-08-04T17:27:01","date_gmt":"2013-08-04T08:27:01","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=865"},"modified":"2021-09-23T09:32:44","modified_gmt":"2021-09-23T00:32:44","slug":"thr201302","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/thr201302\/","title":{"rendered":"\u6771\u5317\u5927\u7406\u7cfb2013\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\u56db\u9762\u4f53 OABC \u306b\u304a\u3044\u3066, \\(\\text{OA} = \\text{OB} = \\text{OC} = 1\\) \u3068\u3059\u308b. \\(\\angle \\text{AOB} = 60^{\\circ}\\) , \\(\\angle \\text{BOC} = 45^{\\circ}\\) , \\(\\angle \\text{COA} = 45^{\\circ}\\) \u3068\u3057, \\(\\overrightarrow{a} = \\overrightarrow{\\text{OA}}\\) , \\(\\overrightarrow{b} = \\overrightarrow{\\text{OB}}\\) , \\(\\overrightarrow{c} = \\overrightarrow{\\text{OC}}\\) \u3068\u304a\u304f.\r\n\u70b9 C \u304b\u3089\u9762 OAB \u306b\u5782\u7dda\u3092\u5f15\u304d, \u305d\u306e\u4ea4\u70b9\u3092 H \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u30d9\u30af\u30c8\u30eb \\(\\overrightarrow{\\text{OH}}\\) \u3092 \\(\\overrightarrow{a}\\) \u3068 \\(\\overrightarrow{b}\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000CH \u306e\u9577\u3055\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u56db\u9762\u4f53 OABC \u306e\u4f53\u7a4d\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

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

    \u6761\u4ef6\u3088\u308a\r\n\\[\r\n\\left\\{\\begin{array}{l} \\left| \\overrightarrow{a} \\right| = \\left| \\overrightarrow{b} \\right| = \\left| \\overrightarrow{c} \\right| = 1 \\\\ \\overrightarrow{a} \\cdot \\overrightarrow{b} = 1 \\cdot 1 \\cos 60^{\\circ} = \\dfrac{1}{2} \\\\ \\overrightarrow{b} \\cdot \\overrightarrow{c} = \\overrightarrow{c} \\cdot \\overrightarrow{a} = 1 \\cdot 1 \\cos 45^{\\circ} = \\dfrac{\\sqrt{2}}{2} \\end{array}\\right. \\ .\r\n\\]\r\n\u70b9 H \u306f\u9762 OAB \u4e0a\u306e\u70b9\u306a\u306e\u3067, \u5b9f\u6570 \\(x , y\\) \u3092\u7528\u3044\u3066\r\n\\[\r\n\\overrightarrow{\\text{OH}} = x \\overrightarrow{a} +y \\overrightarrow{b} \\ .\r\n\\]\r\n\u3068\u304a\u3051\u308b.
    \r\n\u307e\u305f, CH \u306f\u5e73\u9762 OAB \u3068\u5782\u76f4\u306a\u306e\u3067, \\(\\overrightarrow{\\text{CH}} \\perp \\overrightarrow{a}\\) , \\(\\overrightarrow{\\text{CH}} \\perp \\overrightarrow{b}\\) \u3088\u308a\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{CH}} \\cdot \\overrightarrow{a} & = \\left( x \\overrightarrow{a} +y \\overrightarrow{b} -\\overrightarrow{c} \\right) \\cdot \\overrightarrow{a} \\\\\r\n& = x +\\dfrac{y}{2} -\\dfrac{\\sqrt{2}}{2} = 0 \\quad ... [1] \\ , \\\\\r\n\\overrightarrow{\\text{CH}} \\cdot \\overrightarrow{b} & = \\left( x \\overrightarrow{a} +y \\overrightarrow{b} -\\overrightarrow{c} \\right) \\cdot \\overrightarrow{b} \\\\\r\n& = \\dfrac{x}{2} +y -\\dfrac{\\sqrt{2}}{2} = 0 \\quad ... [2] \\ .\r\n\\end{align}\\]\r\n[1] [2] \u3092\u3068\u304f\u3068\r\n\\[\r\nx = y = \\dfrac{\\sqrt{2}}{3} \\ .\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\overrightarrow{\\text{OH}} = \\underline{\\dfrac{\\sqrt{2}}{3} \\left( \\overrightarrow{a} +\\overrightarrow{b} \\right)} \\ .\r\n\\]\r\n

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

    \\[\\begin{align}\r\n\\left| \\overrightarrow{\\text{CH}} \\right|^2 & = \\left| \\dfrac{\\sqrt{2}}{3} \\overrightarrow{a} +\\dfrac{\\sqrt{2}}{3} \\overrightarrow{b} -\\overrightarrow{c} \\right|^2 \\\\\r\n& = \\left( 2 \\cdot \\dfrac{2}{9} +1 \\right) \\cdot 1 +2 \\cdot \\dfrac{2}{9} \\cdot \\dfrac{1}{2} -4 \\cdot \\dfrac{\\sqrt{2}}{3} \\cdot \\dfrac{\\sqrt{2}}{2} \\\\\r\n& = \\dfrac{13}{9} +\\dfrac{2}{9} -\\dfrac{4}{3} \\\\\r\n& = \\dfrac{1}{3} \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\text{CH} = \\underline{\\dfrac{\\sqrt{3}}{3}} \\ .\r\n\\]\r\n

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

    \\[\r\n\\triangle \\text{OAB} = \\dfrac{1}{2} \\cdot 1^2 \\sin 60^{\\circ} = \\dfrac{\\sqrt{3}}{4} \\ .\r\n\\]\r\n\u306a\u306e\u3067, \u6c42\u3081\u308b\u4f53\u7a4d \\(V\\) \u306f\r\n\\[\r\nV = \\dfrac{1}{3} \\cdot \\dfrac{\\sqrt{3}}{4} \\cdot \\dfrac{\\sqrt{3}}{3} = \\underline{\\dfrac{1}{12}} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u56db\u9762\u4f53 OABC \u306b\u304a\u3044\u3066, \\(\\text{OA} = \\text{OB} = \\text{OC} = 1\\) \u3068\u3059\u308b. \\(\\angle \\text{AOB} = 60^{\\circ}\\) , \\(\\angle \\t … \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":[96],"tags":[148,111],"class_list":["post-865","post","type-post","status-publish","format-standard","hentry","category-tohoku_r_2013","tag-tohoku_r","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/865","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=865"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/865\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=865"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=865"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=865"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}