{"id":1986,"date":"2021-11-21T15:56:38","date_gmt":"2021-11-21T06:56:38","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1986"},"modified":"2021-11-21T15:56:38","modified_gmt":"2021-11-21T06:56:38","slug":"osr202102","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/osr202102\/","title":{"rendered":"\u962a\u5927\u7406\u7cfb2021\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\u7a7a\u9593\u5185\u306b, \u540c\u4e00\u5e73\u9762\u4e0a\u306b\u306a\u3044 \\(4\\) \u70b9 O, A, B, C \u304c\u3042\u308b.\r\n\\(s , t \\) \u3092 \\(0 \\lt s \\lt 1\\) , \\(0 \\lt t \\lt 1\\) \u3092\u307f\u305f\u3059\u5b9f\u6570\u3068\u3059\u308b.\r\n\u7dda\u5206 OA \u3092 \\(1 : 1\\) \u306b\u5185\u5206\u3059\u308b\u70b9\u3092 \\(\\text{A} {} _0\\) ,\r\n\u7dda\u5206 OB \u3092 \\(1 : 2\\) \u306b\u5185\u5206\u3059\u308b\u70b9\u3092 \\(\\text{B} {} _0\\) ,\r\n\u7dda\u5206 AC \u3092 \\(s : (1-s)\\) \u306b\u5185\u5206\u3059\u308b\u70b9\u3092 P , \u7dda\u5206 BC \u3092 \\(t : (1-t)\\) \u306b\u5185\u5206\u3059\u308b\u70b9\u3092 Q \u3068\u3059\u308b.\r\n\u3055\u3089\u306b \\(4\\) \u70b9 \\(\\text{A} _0 , \\text{B} {} _0 , \\text{P} , \\text{Q}\\) \u304c\u540c\u4e00\u5e73\u9762\u4e0a\u306b\u3042\u308b\u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(t\\) \u3092 \\(s\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(\\left| \\overrightarrow{\\text{OA}} \\right| = 1\\) ,\r\n\\(\\left| \\overrightarrow{\\text{OB}} \\right| = \\left| \\overrightarrow{\\text{OC}} \\right| = 2\\) ,\r\n\\(\\angle \\text{AOB} = 120^\\circ\\) , \\(\\angle \\text{BOC} = 90^\\circ\\) ,\r\n\\(\\angle \\text{COA} = 60^\\circ\\) , \\(\\angle \\text{POQ} = 90^\\circ\\) \u3067\u3042\u308b\u3068\u304d,\r\n\\(s\\) \u306e\u5024\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

    \\(\\overrightarrow{a} = \\overrightarrow{\\text{OA}}\\) , \\(\\overrightarrow{b} = \\overrightarrow{\\text{OB}}\\) , \\(\\overrightarrow{c} = \\overrightarrow{\\text{OC}}\\) \u3068\u304a\u304f.\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{OA}{} _ 0} & = \\dfrac{1}{2} \\overrightarrow{a} \\ , \\ \\overrightarrow{\\text{OB}{} _ 0} = \\dfrac{1}{3} \\overrightarrow{b} \\ , \\\\\r\n\\overrightarrow{\\text{OP}} & = (1-s) \\overrightarrow{a} +s \\overrightarrow{c}\\ , \\\\\r\n\\overrightarrow{\\text{OQ}} & = (1-t) \\overrightarrow{a} +t \\overrightarrow{c}\r\n\\end{align}\\]\r\n\\(4\\) \u70b9\u304c\u540c\u4e00\u5e73\u9762\u4e0a\u306b\u3042\u308b\u306e\u3067, \\(\\alpha , \\beta\\) \u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{OQ}} & = \\alpha \\overrightarrow{\\text{OP}} +\\beta \\overrightarrow{\\text{OA}{} _ 0} +( 1 -\\alpha -\\beta ) \\overrightarrow{\\text{OB}{} _ 0} \\\\\r\n& = \\left\\{ \\alpha (1-s) +\\dfrac{\\beta}{2} \\right\\}\\overrightarrow{a} +\\dfrac{1 -\\alpha -\\beta}{3} \\overrightarrow{b} +\\alpha s \\overrightarrow{c}\r\n\\end{align}\\]\r\n\u3068\u8868\u305b\u308b.
    \r\n\\(\\overrightarrow{a} , \\overrightarrow{b} , \\overrightarrow{c}\\) \u306f\u4e00\u6b21\u72ec\u7acb\u306a\u306e\u3067\r\n\\[\r\n\\left\\{ \\begin{array}{ll} \\alpha (1-s) +\\dfrac{\\beta}{2} = 0 & ... [1] \\\\ \\dfrac{1 -\\alpha -\\beta}{3} = 1-t & ... [2] \\\\ \\alpha s = t & ... [3] \\end{array} \\right.\r\n\\]\r\n[3] \u3088\u308a, \\(\\alpha = \\dfrac{t}{s}\\) , [2] \u3088\u308a\r\n\\[\r\n\\beta = 1 -\\dfrac{t}{s} -3(1-t) = 3t -\\dfrac{t}{s} -2\r\n\\]\r\n\u3053\u308c\u3089\u3092 [1] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\n\\dfrac{t}{s} (1-s) +\\dfrac{3t}{2} -\\dfrac{t}{2s} -1 & = 0 \\\\\r\n2t -2ts +3ts -t -2s & = 0 \\\\\r\nt (1+s) & = 2s \\\\\r\n\\text{\u2234} \\quad t = \\underline{\\dfrac{2s}{s+1}} &\r\n\\end{align}\\]\r\n

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

    \u6761\u4ef6\u3088\u308a\r\n\\[\\begin{align}\r\n& \\left| \\overrightarrow{a} \\right| = 1 \\ , \\ \\left| \\overrightarrow{b} \\right| = \\left| \\overrightarrow{c} \\right| = 2 \\ , \\\\\r\n& \\overrightarrow{a} \\cdot \\overrightarrow{b} = 1 \\cdot 2 \\cdot \\cos 120^{\\circ} = -1 \\ , \\\\\r\n& \\overrightarrow{b} \\cdot \\overrightarrow{c} = 0 \\ , \\\\\r\n& \\overrightarrow{c} \\cdot \\overrightarrow{a} = 2 \\cdot 1 \\cdot \\cos 60^{\\circ} = 1\r\n\\end{align}\\]\r\n\\(\\text{OP} \\perp \\text{OQ}\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{OP}} \\cdot \\overrightarrow{\\text{OQ}} & = \\left\\{ (1-s) \\overrightarrow{a} +s \\overrightarrow{c} \\right\\} \\cdot \\left\\{ (1-t) \\overrightarrow{b} +t \\overrightarrow{c} \\right\\} \\\\\r\n& = -(1-s) (1-t) +(1-s) t +4st \\\\\r\n& = 2 (s+1) t +s -1 = 0\r\n\\end{align}\\]\r\n(1)<\/strong> \u306e\u7d50\u679c\u3092\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\n2 (s+1) \\cdot \\dfrac{2s}{s+1} +s -1 & = 0 \\\\\r\n5s -1 & = 0 \\\\\r\n\\text{\u2234} \\quad s & = \\underline{\\dfrac{1}{5}}\r\n\\end{align}\\]\r\n\u3053\u308c\u306f \\(0 \\lt s \\lt 1\\) \u3092\u307f\u305f\u3057, \u3053\u306e\u3068\u304d \\(b = \\dfrac{\\frac{2}{5}}{\\frac{1}{5} +1} = \\dfrac{1}{3}\\) \u3067 \\(0 \\lt b \\lt 1\\) \u3082\u307f\u305f\u3057\u3066\u3044\u308b.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u7a7a\u9593\u5185\u306b, \u540c\u4e00\u5e73\u9762\u4e0a\u306b\u306a\u3044 \\(4\\) \u70b9 O, A, B, C \u304c\u3042\u308b. \\(s , t \\) \u3092 \\(0 \\lt s \\lt 1\\) , \\(0 \\lt t \\lt 1\\) \u3092\u307f\u305f\u3059\u5b9f\u6570\u3068\u3059\u308b. \u7dda\u5206 OA \u3092 \\ … \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":[172],"tags":[142,165],"class_list":["post-1986","post","type-post","status-publish","format-standard","hentry","category-osaka_r_2021","tag-osaka_r","tag-165"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1986","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=1986"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1986\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1986"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1986"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1986"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}