{"id":85,"date":"2011-11-26T23:09:35","date_gmt":"2011-11-26T14:09:35","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=85"},"modified":"2021-10-20T15:19:55","modified_gmt":"2021-10-20T06:19:55","slug":"ykr201103","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ykr201103\/","title":{"rendered":"\u6a2a\u56fd\u5927\u7406\u7cfb2011\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\\(1\\) \u8fba\u306e\u9577\u3055\u304c \\(1\\) \u306e\u6b63\u56db\u9762\u4f53 OABC \u306b\u304a\u3044\u3066, \\(3\\) \u8fba OA , OB , AC \u4e0a\u306b\u305d\u308c\u305e\u308c\u70b9 D , E , F \u3092 \\(\\text{OD} = \\dfrac{1}{2}\\) , \\(\\text{OE} = t \\ ( 0 \\lt t \\lt 1 )\\) , \\(\\text{AF} = \\dfrac{2}{3}\\) \u3068\u306a\u308b\u3088\u3046\u306b\u3068\u308b.\r\n\\(\\overrightarrow{\\text{OA}} = \\overrightarrow{a}\\) , \\(\\overrightarrow{\\text{OB}} = \\overrightarrow{b}\\) , \\(\\overrightarrow{\\text{OC}} = \\overrightarrow{c}\\) \u3068\u304a\u304f\u3068\u304d, \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(\\overrightarrow{\\text{DE}} , \\overrightarrow{\\text{DF}}\\) \u3092 \\(\\overrightarrow{a} , \\overrightarrow{b} , \\overrightarrow{c} , t\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(\\overrightarrow{\\text{DE}} \\perp \\overrightarrow{\\text{DF}}\\) \u306e\u3068\u304d, \\(t\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(3\\) \u70b9 D , E , F \u304c\u5b9a\u3081\u308b\u5e73\u9762\u304c\u76f4\u7dda BC \u3068\u4ea4\u308f\u308b\u70b9\u3092 G \u3068\u3059\u308b\u3068\u304d, \u7dda\u5206 BG \u306e\u9577\u3055\u3092 \\(t\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/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\\[\\begin{align}\r\n\\overrightarrow{\\text{OD}} & = \\dfrac{1}{2} \\overrightarrow{a} , \\ \\overrightarrow{\\text{OE}} = t \\overrightarrow{b} , \\\\\r\n\\overrightarrow{\\text{OF}} & = \\dfrac{\\overrightarrow{a} +2 \\overrightarrow{c}}{3}\r\n\\end{align}\\]\r\n\u3086\u3048\u306b\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{DE}} & = \\overrightarrow{\\text{OE}} -\\overrightarrow{\\text{OD}} = \\underline{-\\dfrac{1}{2} \\overrightarrow{a} +t \\overrightarrow{b}} , \\\\\r\n\\overrightarrow{\\text{DF}} & = \\overrightarrow{\\text{OF}} -\\overrightarrow{\\text{OD}} = \\underline{-\\dfrac{1}{6} \\overrightarrow{a} +\\dfrac{2}{3} \\overrightarrow{c}}\r\n\\end{align}\\]\r\n

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

    \u6761\u4ef6\u3088\u308a\r\n\\[\\begin{align}\r\n\\big| \\overrightarrow{a} \\big| & = \\big| \\overrightarrow{b} \\big| = \\big| \\overrightarrow{c} \\big| = 1 , \\\\\r\n\\overrightarrow{a} \\cdot \\overrightarrow{b} & = \\overrightarrow{b} \\cdot \\overrightarrow{c} = \\cdot \\overrightarrow{c} \\cdot \\overrightarrow{a} = 1 \\cdot 1 \\cdot \\cos \\dfrac{\\pi}{3} = \\dfrac{1}{2}\r\n\\end{align}\\]\r\n\\(\\overrightarrow{\\text{DE}} \\perp \\overrightarrow{\\text{DF}}\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{DE}} \\cdot \\overrightarrow{\\text{DF}} & = \\left( -\\dfrac{1}{2} \\overrightarrow{a} +t \\overrightarrow{b} \\right) \\cdot \\left( -\\dfrac{1}{6} \\overrightarrow{a} +\\dfrac{2}{3} \\overrightarrow{c} \\right) \\\\\r\n& = \\dfrac{1}{12} \\big| \\overrightarrow{a} \\big|^2 -\\dfrac{1}{3} \\overrightarrow{a} \\cdot \\overrightarrow{c} -\\dfrac{t}{6} \\overrightarrow{a} \\cdot \\overrightarrow{b} +\\dfrac{2t}{3} \\overrightarrow{b} \\cdot \\overrightarrow{c} \\\\\r\n& = \\dfrac{t}{4} -\\dfrac{1}{12} = 0 \\\\\r\n\\text{\u2234} \\quad t & = \\underline{\\dfrac{1}{3}}\r\n\\end{align}\\]\r\n

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

    \u70b9 G \u306f\u5e73\u9762 DEF \u4e0a\u306b\u3042\u308b\u306e\u3067, \u5b9f\u6570 \\(u , v\\) \u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{OG}} & = \\overrightarrow{\\text{OD}} +u \\overrightarrow{\\text{DE}} +v \\overrightarrow{\\text{DF}} \\\\\r\n& = \\dfrac{1}{2} \\overrightarrow{a} +u \\left( -\\dfrac{1}{2} \\overrightarrow{a} +t \\overrightarrow{b} \\right) +v \\left( -\\dfrac{1}{6} \\overrightarrow{a} +\\dfrac{2}{3} \\overrightarrow{c} \\right) \\\\\r\n& = \\dfrac{3-3u-v}{6} \\overrightarrow{a} +t u \\overrightarrow{b} +\\dfrac{2v}{3} \\overrightarrow{c}\r\n\\end{align}\\]\r\n\u3068\u8868\u305b\u308b.
    \r\n\u3055\u3089\u306b, \u70b9 G \u306f\u8fba BC \u4e0a\u306b\u3042\u308b\u3053\u3068\u304b\u3089\r\n\\[\\begin{align}\r\n\\dfrac{3-3u-v}{6} = 0 & , \\ ut +\\dfrac{2v}{3} = 1 \\\\\r\n3u+v = 3 & , \\ 3tu+2v = 3 \\\\\r\n\\text{\u2234} \\quad u = \\dfrac{1}{2-t} & , \\ v = \\dfrac{3(1-t)}{2-t}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066,\r\n\\[\r\n\\overrightarrow{\\text{OG}} = \\dfrac{t}{2-t} \\overrightarrow{b} +\\dfrac{2(1-t)}{2-t} \\overrightarrow{c}\r\n\\]\r\n\u3088\u3063\u3066, \u70b9 G \u306f\u8fba BC \u3092 \\(2(1-t) : t\\) \u306b\u5185\u5206\u3059\u308b\u306e\u3067\r\n\\[\r\n\\text{BG} = \\underline{\\dfrac{2(1-t)}{2-t}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(1\\) \u8fba\u306e\u9577\u3055\u304c \\(1\\) \u306e\u6b63\u56db\u9762\u4f53 OABC \u306b\u304a\u3044\u3066, \\(3\\) \u8fba OA , OB , AC \u4e0a\u306b\u305d\u308c\u305e\u308c\u70b9 D , E , F \u3092 \\(\\text{OD} = \\dfrac{1}{2}\\) , \\(\\ … \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":[44],"tags":[13,9],"class_list":["post-85","post","type-post","status-publish","format-standard","hentry","category-yokokoku_r_2011","tag-13","tag-yokokoku_r"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/85","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=85"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/85\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=85"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=85"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=85"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}