{"id":488,"date":"2012-11-25T09:33:37","date_gmt":"2012-11-25T00:33:37","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=488"},"modified":"2021-09-23T22:27:11","modified_gmt":"2021-09-23T13:27:11","slug":"thr201104","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/thr201104\/","title":{"rendered":"\u6771\u5317\u5927\u7406\u7cfb2011\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\u5e73\u9762\u4e0a\u306b\u9577\u3055 \\(3\\) \u306e\u7dda\u5206 OA \u3092\u8003\u3048, \u30d9\u30af\u30c8\u30eb \\(\\overrightarrow{\\text{OA}}\\) \u3092 \\(\\overrightarrow{a}\\) \u3067\u8868\u3059. \\(0 \\lt t \\lt 1\\) \u3092\u6e80\u305f\u3059\u5b9f\u6570 \\(t\\) \u306b\u5bfe\u3057\u3066, \\(\\overrightarrow{\\text{OP}} = t \\overrightarrow{a}\\) \u3068\u306a\u308b\u3088\u3046\u306b\u70b9 P \u3092\u5b9a\u3081\u308b.\r\n\u5927\u304d\u3055 \\(2\\) \u306e\u30d9\u30af\u30c8\u30eb \\(\\overrightarrow{b}\\) \u3092 \\(\\overrightarrow{a}\\) \u3068\u89d2 \\(\\theta \\ ( 0 \\lt \\theta \\lt \\pi )\\) \u3092\u306a\u3059\u3088\u3046\u306b\u3068\u308a, \u70b9 B \u3092 \\(\\overrightarrow{\\text{OB}} = \\overrightarrow{b}\\) \u3067\u5b9a\u3081\u308b.\r\n\u7dda\u5206 OB \u306e\u4e2d\u70b9\u3092 Q \u3068\u3057, \u7dda\u5206 AQ \u3068\u7dda\u5206 BP \u306e\u4ea4\u70b9\u3092 R \u3068\u3059\u308b.
\r\n\u3000\u3053\u306e\u3068\u304d, \u3069\u306e\u3088\u3046\u306b \\(\\theta\\) \u3092\u3068\u3063\u3066\u3082 \\(\\overrightarrow{\\text{OR}}\\) \u3068 \\(\\overrightarrow{\\text{AB}}\\) \u304c\u5782\u76f4\u306b\u306a\u3089\u306a\u3044\u3088\u3046\u306a \\(t\\) \u306e\u5024\u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p>\r\n

\r\n\r\n

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

\\(\\overrightarrow{\\text{OA}} = \\overrightarrow{a}\\) , \\(\\overrightarrow{\\text{OB}} = \\overrightarrow{b}\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\left| \\overrightarrow{a} \\right| = 3 , \\ \\left| \\overrightarrow{b} \\right| = 2 , \\ \\overrightarrow{a} \\cdot \\overrightarrow{b} = 6 \\cos \\theta \\ .\r\n\\]\r\n\u30e1\u30cd\u30e9\u30a6\u30b9\u306e\u5b9a\u7406\u3092\u7528\u3044\u308b\u3068\r\n\\[\\begin{align}\r\n\\dfrac{\\text{AP}}{\\text{PO}} \\cdot \\dfrac{\\text{OB}}{\\text{BQ}} \\cdot \\dfrac{\\text{QR}}{\\text{RA}} & = 1\\\\\r\n\\dfrac{1-t}{t} \\cdot \\dfrac{2}{1} \\cdot \\dfrac{\\text{QR}}{\\text{RA}} & = 1 \\\\\r\n\\text{\u2234} \\quad \\dfrac{\\text{QR}}{\\text{RA}} = \\dfrac{t}{2(1-t)} & \\ .\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{OR}} & = \\dfrac{t \\overrightarrow{\\text{OA}} +2(1-t) \\overrightarrow{\\text{OQ}}}{t +2(1-t)} \\\\\r\n& = \\dfrac{t \\overrightarrow{a} +(1-t) \\overrightarrow{b}}{2-t} \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{OR}} \\cdot \\overrightarrow{\\text{AB}} & = \\dfrac{1}{2-t} \\left\\{ t \\overrightarrow{a} +(1-t) \\overrightarrow{b} \\right\\} \\cdot \\left( \\overrightarrow{b} -\\overrightarrow{a} \\right) \\\\\r\n& = \\dfrac{1}{2-t} \\left\\{ (1-t) \\left| \\overrightarrow{b} \\right|^2 -t \\left| \\overrightarrow{a} \\right|^2 +(2t-1) \\overrightarrow{a} \\cdot \\overrightarrow{b} \\right\\} \\\\\r\n& = \\dfrac{1}{2-t} \\left\\{ 4(1-t) -9t +6(2t-1) \\cos \\theta \\right\\} \\\\\r\n& = \\dfrac{1}{2-t} \\underline{\\left\\{ 4-13t +6(2t-1) \\cos \\theta \\right\\}} _ {\\text{[A]}} \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u4e0b\u7dda\u90e8 [A] \u304c \\(0\\) \u306b\u306a\u3089\u306a\u3044\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.
\r\n\\(u =\\cos \\theta\\) \u3068\u304a\u304f\u3068, \\(0 \\lt \\theta \\lt \\pi\\) \u3088\u308a\r\n\\[\r\n-1 \\lt u \\lt 1 \\ .\r\n\\]\r\n\u4e0b\u7dda\u90e8 [A] \u3092 \\(u\\) \u306e\u95a2\u6570\u3068\u307f\u306a\u3057\u3066 \\(f(u)\\) \u3068\u304a\u304f\u3068, \u3053\u308c\u306f\u9ad8\u3005 \\(1\\) \u6b21\u95a2\u6570\u3067\u3042\u308b.\r\n\\[\\begin{align}\r\nf(1) & = 4-13t +6(2t-1) \\\\\r\n& = -2-t \\lt 0 \\quad ( \\ \\text{\u2235} \\ 0 \\lt t \\lt 1 \\ ) \\ .\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \u6c42\u3081\u308b\u6761\u4ef6\u306f\r\n\\[\\begin{align}\r\nf(-1) & \\leqq 0 \\\\\r\n4-13t -6(2t-1) & \\leqq 0 \\\\\r\n10-25t & \\leqq 0 \\\\\r\n\\text{\u2234} \\quad t & \\geqq \\dfrac{2}{5} \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\(0 \\lt t \\lt 1\\) \u3068\u3042\u308f\u305b\u3066\r\n\\[\r\n\\underline{\\dfrac{2}{5} \\leqq t \\lt 1} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u5e73\u9762\u4e0a\u306b\u9577\u3055 \\(3\\) \u306e\u7dda\u5206 OA \u3092\u8003\u3048, \u30d9\u30af\u30c8\u30eb \\(\\overrightarrow{\\text{OA}}\\) \u3092 \\(\\overrightarrow{a}\\) \u3067\u8868\u3059. \\(0 \\lt t \\lt 1\\) \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":[73],"tags":[148,13],"class_list":["post-488","post","type-post","status-publish","format-standard","hentry","category-tohoku_r_2011","tag-tohoku_r","tag-13"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/488","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=488"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/488\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=488"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=488"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=488"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}