\\(S\\) \u3092, \u5ea7\u6a19\u7a7a\u9593\u5185\u306e\u539f\u70b9 O \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(1\\) \u306e\u7403\u9762\u3068\u3059\u308b.
\n\\(S\\) \u4e0a\u3092\u52d5\u304f\u70b9 A, B, C, D \u306b\u5bfe\u3057\u3066
\n\\[
\nF = 2 ( \\text{AB}^2 + \\text{BC}^2 + \\text{CA}^2 ) -3 ( \\text{AD}^2 + \\text{BD}^2 + \\text{CD}^2 )
\n\\]\n\u3068\u304a\u304f. \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\n
(1)<\/strong>\u3000\\(\\overrightarrow{\\text{OA}} = \\overrightarrow{a}\\) , \\(\\overrightarrow{\\text{OB}} = \\overrightarrow{b}\\) , \\(\\overrightarrow{\\text{OC}} = \\overrightarrow{c}\\) , \\(\\overrightarrow{\\text{OD}} = \\overrightarrow{d}\\) \u3068\u3059\u308b\u3068\u304d, \\(\\overrightarrow{a} , \\overrightarrow{b} , \\overrightarrow{c} , \\overrightarrow{d}\\) \u306b\u3088\u3089\u306a\u3044\u5b9a\u6570 \\(k\\) \u306b\u3088\u3063\u3066 (2)<\/strong>\u3000\u70b9 A, B, C, D \u304c\u7403\u9762 \\(S\\) \u4e0a\u3092\u52d5\u304f\u3068\u304d\u306e, \\(F\\) \u306e\u6700\u5927\u5024 \\(M\\) \u3092\u6c42\u3081\u3088.<\/p>\n<\/li>\n (3)<\/strong>\u3000\u70b9 C \u306e\u5ea7\u6a19\u304c \\(\\left( -\\dfrac{1}{4} , \\dfrac{\\sqrt{15}}{4} , 0 \\right)\\) , \u70b9 D \u306e\u5ea7\u6a19\u304c \\(( 1 , 0 , 0 )\\) \u3067\u3042\u308b\u3068\u304d, \\(F = M\\) \u3068\u306a\u308b \\(S\\) \u4e0a\u306e\u70b9 A, B \u306e\u7d44\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p>\n<\/li>\n<\/ol>\n (1)<\/strong><\/p>\n \\(\\left| \\overrightarrow{a} \\right| = \\left| \\overrightarrow{b} \\right| = \\left| \\overrightarrow{c} \\right| = \\left| \\overrightarrow{d} \\right| = 1\\) \u3067\u3042\u308b\u3053\u3068\u3092\u7528\u3044\u308c\u3070 (2)<\/strong><\/p>\n \\(\\overrightarrow{\\text{OG}} = \\overrightarrow{g} = \\dfrac{\\overrightarrow{a} +\\overrightarrow{b} +\\overrightarrow{c}}{3}\\) \u3068\u304a\u304f\u3068 (3)<\/strong><\/p>\n D \u306e\u5ea7\u6a19\u304b\u3089, G \\(\\left( \\dfrac{1}{2} , 0 , 0 \\right)\\) . \\(S\\) \u3092, \u5ea7\u6a19\u7a7a\u9593\u5185\u306e\u539f\u70b9 O \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(1\\) \u306e\u7403\u9762\u3068\u3059\u308b. \\(S\\) \u4e0a\u3092\u52d5\u304f\u70b9 A, B, C, D \u306b\u5bfe\u3057\u3066 \\[ F = 2 ( \\text{AB}^2 + \\text{BC}^2 + […]<\/p>\n","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":[3],"tags":[],"class_list":["post-59","post","type-post","status-publish","format-standard","hentry","category-web","entry"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/log\/wp-json\/wp\/v2\/posts\/59","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/log\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/log\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/log\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/log\/wp-json\/wp\/v2\/comments?post=59"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/log\/wp-json\/wp\/v2\/posts\/59\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/log\/wp-json\/wp\/v2\/media?parent=59"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/log\/wp-json\/wp\/v2\/categories?post=59"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/log\/wp-json\/wp\/v2\/tags?post=59"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\\[
\nF = k \\left( \\overrightarrow{a} +\\overrightarrow{b} +\\overrightarrow{c} \\right) \\cdot \\left( \\overrightarrow{a} +\\overrightarrow{b} +\\overrightarrow{c} -3 \\overrightarrow{d} \\right)
\n\\]\n\u3068\u66f8\u3051\u308b\u3053\u3068\u3092\u793a\u3057, \u5b9a\u6570 \\(k\\) \u3092\u6c42\u3081\u3088.<\/p>\n<\/li>\n
\n
\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\n
\n\\[\\begin{align}
\nF & = 2 \\left\\{ 6 -2 \\left( \\overrightarrow{a} \\cdot \\overrightarrow{b} +\\overrightarrow{b} \\cdot \\overrightarrow{c} +\\overrightarrow{c} \\cdot \\overrightarrow{a} \\right) \\right\\} \\\\
\n& \\qquad -3 \\left\\{ 6 -2 \\left( \\overrightarrow{a} +\\overrightarrow{b} + \\overrightarrow{c} \\right) \\cdot \\overrightarrow{d} \\right\\} \\\\
\n& = -6 -4 \\left( \\overrightarrow{a} \\cdot \\overrightarrow{b} +\\overrightarrow{b} \\cdot \\overrightarrow{c} +\\overrightarrow{c} \\cdot \\overrightarrow{a} \\right) \\\\
\n& \\qquad +6 \\left( \\overrightarrow{a} +\\overrightarrow{b} + \\overrightarrow{c} \\right) \\cdot \\overrightarrow{d}
\n\\end{align}\\]\n\\(P = \\left( \\overrightarrow{a} +\\overrightarrow{b} +\\overrightarrow{c} \\right) \\cdot \\left( \\overrightarrow{a} +\\overrightarrow{b} +\\overrightarrow{c} -3 \\overrightarrow{d} \\right)\\) \u3068\u304a\u304f\u3068
\n\\[\\begin{align}
\nP & = 3 +2 \\left( \\overrightarrow{a} \\cdot \\overrightarrow{b} +\\overrightarrow{b} \\cdot \\overrightarrow{c} +\\overrightarrow{c} \\cdot \\overrightarrow{a} \\right) \\\\
\n& \\qquad -3 \\left( \\overrightarrow{a} +\\overrightarrow{b} + \\overrightarrow{c} \\right) \\cdot \\overrightarrow{d}
\n\\end{align}\\]\n\u3088\u3063\u3066
\n\\[
\nF = -2 P
\n\\]\n\u3068\u66f8\u3051\u3066
\n\\[
\nk = \\underline{-2}
\n\\]\n
\n\\[\\begin{align}
\nF & = -18 \\overrightarrow{g} \\cdot \\left( \\overrightarrow{g} -\\overrightarrow{d} \\right) \\\\
\n& = -18 \\left| \\overrightarrow{g} -\\dfrac{1}{2} \\overrightarrow{d} \\right|^2 +\\dfrac{9}{2} \\quad \\left( \\ \\text{\u2235} \\ \\left| \\overrightarrow{d} \\right| = 1 \\ \\right) \\\\
\n& \\leqq \\dfrac{9}{2}
\n\\end{align}\\]\n\u7b49\u53f7\u6210\u7acb\u306f \\(\\overrightarrow{g} = \\dfrac{1}{2} \\overrightarrow{d}\\) \u306e\u3068\u304d\u3067\u3042\u308b\u304c,
\n\u70b9 G \u306f \\(\\triangle \\text{ABC}\\) \u306e\u91cd\u5fc3\u306a\u306e\u3067 \\(S\\) \u306e\u5185\u5074\u306b\u3042\u308a, \\(\\text{OG} = \\dfrac{1}{2}\\) \u3068\u306a\u308b\u3088\u3046\u306b A, B, C \u3092\u3068\u3063\u3066, \u534a\u76f4\u7dda OG \u4e0a\u306b D \u3092\u3068\u308c\u3070, \u7b49\u53f7\u304c\u6210\u7acb\u3059\u308b.
\n\u3088\u3063\u3066
\n\\[
\nM = \\underline{\\dfrac{9}{2}}
\n\\]\n
\n\\(\\overrightarrow{\\text{OM}} = \\overrightarrow{m} = \\dfrac{\\overrightarrow{a} +\\overrightarrow{b}}{2}\\) \u3068\u304a\u304f\u3068, M \u306f AB \u306e\u4e2d\u70b9\u3067
\n\\[\\begin{align}
\n\\overrightarrow{g} & = \\dfrac{2 \\overrightarrow{m} +\\overrightarrow{c}}{3} \\\\
\n\\text{\u2234} \\quad \\overrightarrow{m} & = \\dfrac{3 \\overrightarrow{g} -\\overrightarrow{c}}{2} \\\\
\n& = \\dfrac{1}{2} \\left( \\begin{array}{c} \\dfrac{3}{2} +\\dfrac{1}{4} \\\\ \\dfrac{\\sqrt{15}}{4} \\\\ 0 \\end{array} \\right) = \\left( \\begin{array}{c} \\dfrac{7}{8} \\\\ \\dfrac{\\sqrt{15}}{8} \\\\ 0 \\end{array} \\right)
\n\\end{align}\\]\n\u3053\u306e\u3068\u304d, \\(\\left( \\dfrac{7}{8} \\right)^2 +\\left( \\dfrac{\\sqrt{15}}{8} \\right)^2 = 1\\) \u306a\u306e\u3067, M \u306f \\(S\\) \u4e0a\u306b\u3042\u308b.
\nA, B, M \u306f\u3059\u3079\u3066 \\(S\\) \u4e0a\u306b\u3042\u308b\u306e\u3067, \\(S\\) \u306e\u65ad\u9762\u3067\u3042\u308b\u540c\u4e00\u5186\u5468\u4e0a\u306b\u3042\u308b\u3053\u3068\u306b\u306a\u308b\u304c, \u3053\u308c\u306f \\(3\\) \u70b9\u304c\u4e00\u81f4\u3059\u308b\u5834\u5408\u306b\u9650\u3089\u308c\u308b.
\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u70b9\u306e\u7d44\u306f
\n\\[
\n\\underline{\\text{A} \\ \\left( \\dfrac{7}{8} , \\dfrac{\\sqrt{15}}{8} , 0 \\right) \\ , \\ \\text{B} \\ \\left( \\dfrac{7}{8} , \\dfrac{\\sqrt{15}}{8} , 0 \\right)}
\n\\]\n<\/div><\/div>\n