{"id":5455,"date":"2024-11-26T16:59:21","date_gmt":"2024-11-26T08:59:21","guid":{"rendered":"http:\/\/jhwk.online\/?p=3858"},"modified":"2025-12-24T15:06:56","modified_gmt":"2025-12-24T07:06:56","slug":"%e7%a9%ba%e9%97%b4%e5%90%91%e9%87%8f","status":"publish","type":"post","link":"http:\/\/jhwk.online\/?p=5455","title":{"rendered":"\u7a7a\u95f4\u5411\u91cf"},"content":{"rendered":"\n<div class=\"wp-block-ht-block-toc  is-style-outline htoc htoc--position-wide toc-list-style-plain\" data-htoc-state=\"expanded\"><span class=\"htoc__title\"><span class=\"ht_toc_title\">\u76ee\u5f55<\/span><span class=\"htoc__toggle\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16\" height=\"16\"><g fill=\"#444\"><path d=\"M15 7H1c-.6 0-1 .4-1 1s.4 1 1 1h14c.6 0 1-.4 1-1s-.4-1-1-1z\"><\/path><path d=\"M15 1H1c-.6 0-1 .4-1 1s.4 1 1 1h14c.6 0 1-.4 1-1s-.4-1-1-1zM15 13H1c-.6 0-1 .4-1 1s.4 1 1 1h14c.6 0 1-.4 1-1s-.4-1-1-1z\"><\/path><\/g><\/svg><\/span><\/span><div class=\"htoc__itemswrap\"><ul class=\"ht_toc_list\"><li class=\"\"><a href=\"\/#htoc-\">\u7a7a\u95f4\u5411\u91cf\u57fa\u672c\u5b9a\u7406<\/a><\/li><li class=\"\"><a href=\"\/#htoc-11111\">\u7a7a\u95f4\u76f4\u89d2\u5750\u6807\u7cfb<\/a><\/li><\/ul><\/div><\/div>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"htoc-\">\u7a7a\u95f4\u5411\u91cf\u57fa\u672c\u5b9a\u7406<\/h4>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>1.(\u591a\u9009)\u4e0b\u5217\u8bf4\u6cd5\u4e2d\u6b63\u786e\u7684\u662f(\\quad )~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA. \u5df2\u77e5\u7a7a\u95f4\u5411\u91cf  \\boldsymbol{a}, \\boldsymbol{b} ,\u5219 |\\boldsymbol{a}|-|\\boldsymbol{b}|=|\\boldsymbol{a}+\\boldsymbol{b}|  \u662f  \\boldsymbol{a}, \\boldsymbol{b}  \u5171\u7ebf\u7684\u5145\u8981\u6761\u4ef6\uff1b~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nB. \u82e5  \\overrightarrow{A B}, \\overrightarrow{C D}  \u5171\u7ebf, \u5219  A B \/ \/ C D;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\\\\nC.  A, B, C  \u4e09\u70b9\u4e0d\u5171\u7ebf, \u5bf9\u7a7a\u95f4\u4efb\u610f\u4e00\u70b9  O , \u82e5  \\overrightarrow{O P}=\\frac{3}{4} \\overrightarrow{O A}+\\frac{1}{8} \\overrightarrow{O B}+\\frac{1}{8} \\overrightarrow{O C} , \u5219  P, A ,\n B, C\\\\\u56db\u70b9\u5171\u9762;  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nD. \u82e5  P, A, B, C  \u4e3a\u7a7a\u95f4\u56db\u70b9, \u4e14\u6709  \\overrightarrow{P A}=\\lambda \\overrightarrow{P B}+\\mu \\overrightarrow{P C}(\\overrightarrow{P B}, \\overrightarrow{P C}  \u4e0d\u5171\u7ebf  ) , \u5219  \\lambda +\\mu=1 \n\n\u662f~~~~\\\\  A, B, C  \u4e09\u70b9\u5171\u7ebf\u7684\u5145\u8981\u6761\u4ef6.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-cd\"><pre>\u7b54\u6848: CD.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>2. \u5982\u56fe, \u5728\u5e73\u884c\u516d\u9762\u4f53  A B C D-A^{\\prime} B^{\\prime} C^{\\prime} D^{\\prime}  \u4e2d,  A B=2, A D=2, A A^{\\prime}=3, \\angle B A D=~~~~~~~~\\\\\\angle B A A^{\\prime}=   \\angle D A A^{\\prime}=60^{\\circ} . \u6c42 (1)\u6c42 B C^{\\prime}  \u4e0e  C A^{\\prime}  \u7684\u957f;(2) B C^{\\prime}  \u4e0e  C A^{\\prime}  \u7684\u5939\u89d2.~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-7d4818b\" data-block-id=\"7d4818b\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-7d4818b-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-87a8342\" data-v=\"4\" data-block-id=\"87a8342\"><style>@media screen and (min-width:690px){.stk-87a8342 {flex:var(--stk-flex-grow, 1) 1 calc(71.9\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-1-sqrt-19-3-2-90-circ-1-overrightarrow-a-b-boldsymbol-a-overrightarrow-a-d-boldsymbol-b-overrightarrow-a-a-prime-boldsymbol-c-overrightarrow-b-c-prime-boldsymbol-b-boldsymbol-c-overrightarrow-c-a-prime-boldsymbol-a-boldsymbol-b-boldsymbol-c-overrightarrow-bc-prime-sqrt-19-overrightarrow-c-a-prime-3-2-overrightarrow-b-c-prime-cdot-overrightarrow-c-a-prime-boldsymbol-b-boldsymbol-c-cdot-boldsymbol-a-boldsymbol-b-boldsymbol-c-0-b-c-prime-c-a-prime-90-circ\"><pre>\u7b54\u6848:(1)\\sqrt{19},3,(2)90^{\\circ} ;\n\u63d0\u793a:(1)\u8bbe  \\overrightarrow{A B}=\\boldsymbol{a}, \\overrightarrow{A D}=\\boldsymbol{b}, ~~~~~~~\\\\\\overrightarrow{A A^{\\prime}}=\\boldsymbol{c} , \u5219  \\overrightarrow{B C^{\\prime}}=\\boldsymbol{b}+\\boldsymbol{c} ,\\overrightarrow{C A^{\\prime}}=-\\boldsymbol{a}-\\boldsymbol{b}+\\boldsymbol{c} ,~~~~~~~~~~~~~~~~~~~~~~~\\\\|\\overrightarrow{BC^{\\prime}}|=\\sqrt{19},|\\overrightarrow{C A^{\\prime}}|=3.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\(2)\\overrightarrow{B C^{\\prime}} \\cdot \\overrightarrow{C A^{\\prime}}=(\\boldsymbol{b}+\\boldsymbol{c}) \\cdot(-\\boldsymbol{a}-\\boldsymbol{b}+\\boldsymbol{c}) =0~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n B C^{\\prime}  \u4e0e  C A^{\\prime}  \u7684\u5939\u89d2\u4e3a90^{\\circ}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-701d15b\" data-v=\"4\" data-block-id=\"701d15b\"><style>@media screen and (min-width:690px){.stk-701d15b {flex:var(--stk-flex-grow, 1) 1 calc(28.1\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl1.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>3.\u5982\u56fe, \u6b63\u56db\u9762\u4f53  A B C D \u7684\u68f1\u957f\u4e3a  1, E, F ,\n G, H  \u5206\u522b\u662f\u6b63\u56db\u9762\u4f53  A B C D  \u4e2d\u5404\u68f1\u7684\u4e2d\u70b9.~~~\\\\\n\u6c42: (1)  \\overrightarrow{E F}  \u7684\u6a21\u957f; (2)\u6c42  \\overrightarrow{E F} \\cdot  \\overrightarrow{G H} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-6b94ff2\" data-block-id=\"6b94ff2\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-6b94ff2-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-5339914\" data-v=\"4\" data-block-id=\"5339914\"><style>@media screen and (min-width:690px){.stk-5339914 {flex:var(--stk-flex-grow, 1) 1 calc(68\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-1-overrightarrow-e-f-frac-sqrt-2-2-2-0-1-overrightarrow-a-b-boldsymbol-a-overrightarrow-a-c-boldsymbol-b-overrightarrow-a-d-boldsymbol-c-therefore-overrightarrow-e-f-overrightarrow-e-a-overrightarrow-a-f-frac-1-2-boldsymbol-c-boldsymbol-a-boldsymbol-b-therefore-overrightarrow-e-f-frac-sqrt-2-2-2-overrightarrow-e-f-cdot-overrightarrow-g-h-frac-1-2-boldsymbol-c-boldsymbol-a-boldsymbol-b-cdot-frac-1-2-boldsymbol-b-boldsymbol-c-boldsymbol-a-0\"><pre>\u7b54\u6848:(1)|\\overrightarrow{E F}|=\\frac{\\sqrt{2}}{2},(2)0;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\u63d0\u793a:(1)\n\\overrightarrow{A B}=\\boldsymbol{a}, \\overrightarrow{A C}=\\boldsymbol{b}, \\overrightarrow{A D}=\\boldsymbol{c},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\n\n\\therefore  \\overrightarrow{E F}=\\overrightarrow{E A}+\\overrightarrow{A F}\n=\\frac{1}{2}(\\boldsymbol{c}-\\boldsymbol{a}-\\boldsymbol{b}), \n\\therefore  |\\overrightarrow{E F}|=\\frac{\\sqrt{2}}{2} .~~~~~~\\\\(2)\n\n\\overrightarrow{E F} \\cdot \\overrightarrow{G H}=\\frac{1}{2}(\\boldsymbol{c}-\\boldsymbol{a}-\\boldsymbol{b}) \\cdot \\frac{1}{2}(\\boldsymbol{b}+\\boldsymbol{c}-\\boldsymbol{a})=0.~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-1155e73\" data-v=\"4\" data-block-id=\"1155e73\"><style>@media screen and (min-width:690px){.stk-1155e73 {flex:var(--stk-flex-grow, 1) 1 calc(32\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"200\" height=\"168\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl2-2.png\" alt=\"\" class=\"wp-image-3955\" style=\"width:177px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>4.\u5982\u56fe, \u4e8c\u9762\u89d2  \\alpha-l-\\beta  \u7684\u68f1\u4e0a\u6709\u4e24\u4e2a\u70b9  A, B , \u7ebf\u6bb5  B D  \u4e0e  A C  \u5206\u522b\u5728\u8fd9\u4e2a\u4e8c\u9762\u89d2\u7684\u4e24\u4e2a~\\\\\u9762\u5185, \u5e76\u4e14\u90fd\u5782\u76f4\u4e8e\u68f1  l . \u82e5  A B=4, A C=6, B D=8, C D=2 \\sqrt{17} , \u6c42\u5e73\u9762  \\alpha  \u4e0e\u5e73\u9762  \\beta  \u7684~~\\\\\u5939\u89d2.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-45d2f5e\" data-block-id=\"45d2f5e\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-45d2f5e-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-49ae3b5\" data-v=\"4\" data-block-id=\"49ae3b5\"><style>@media screen and (min-width:690px){.stk-49ae3b5 {flex:var(--stk-flex-grow, 1) 1 66.7\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-60-circ-overrightarrow-c-d-overrightarrow-c-a-overrightarrow-a-b-overrightarrow-b-d-overrightarrow-c-d-2-overrightarrow-c-a-overrightarrow-a-b-overrightarrow-b-d-2-rightarrow-cos-langle-overrightarrow-c-a-overrightarrow-b-d-rangle-frac-1-2-therefore-langle-overrightarrow-c-a-overrightarrow-b-d-rangle-120-circ-therefore-text-alpha-text-beta-text-60-circ-text\"><pre>\u7b54\u6848:60^{\\circ} ;\u63d0\u793a:\\overrightarrow{C D}=\\overrightarrow{C A}+\\overrightarrow{A B}+\\overrightarrow{B D},~~~~~~~~~~~~~~~~~~~~\\\\|\\overrightarrow{C D}|^{2}=(\\overrightarrow{C A}+\\overrightarrow{A B}+\\overrightarrow{B D})^{2} \\Rightarrow \\cos \\langle\\overrightarrow{C A}, \\overrightarrow{B D}\\rangle=-\\frac{1}{2},~\n\\\\\\therefore\\langle\\overrightarrow{C A}, \\overrightarrow{B D}\\rangle=120^{\\circ}, \\therefore \\text { \u5e73\u9762 } \\alpha \\text { \u4e0e\u5e73\u9762 } \\beta \\text { \u5939\u89d2\u4e3a } 60^{\\circ} \\text {. }~\n<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-f4e1b7f\" data-v=\"4\" data-block-id=\"f4e1b7f\"><style>@media screen and (min-width:690px){.stk-f4e1b7f {flex:var(--stk-flex-grow, 1) 1 33.3\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl3.png\" alt=\"\" style=\"width:207px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-text-color has-palette-color-2-color has-alpha-channel-opacity has-palette-color-2-background-color has-background is-style-wide\"\/>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"htoc-11111\">\u7a7a\u95f4\u76f4\u89d2\u5750\u6807\u7cfb<\/h4>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>1.\u5982\u56fe, \u5728\u68f1\u957f\u4e3a 1 \u7684\u6b63\u65b9\u4f53  A B C D-A_{1} B_{1} C_{1} D_{1}  \u4e2d,  E  \u4e3a\u7ebf\u6bb5  D D_{1}  \u7684\u4e2d\u70b9,  F  \u4e3a\u7ebf\u6bb5  B B_{1}  \\\\\u7684\u4e2d\u70b9.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(1) \u6c42\u70b9  A_{1}  \u5230\u76f4\u7ebf  B_{1} E  \u7684\u8ddd\u79bb;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\n(2) \u6c42\u70b9  A_{1}  \u5230\u5e73\u9762  A B_{1} E  \u7684\u8ddd\u79bb;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(3) \u6c42\u76f4\u7ebf  F C_{1}  \u5230\u5e73\u9762  A B_{1} E  \u7684\u8ddd\u79bb.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-df45d5e\" data-block-id=\"df45d5e\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-df45d5e-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-e5451b8\" data-v=\"4\" data-block-id=\"e5451b8\"><style>@media screen and (min-width:690px){.stk-e5451b8 {flex:var(--stk-flex-grow, 1) 1 calc(70.5\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-1-frac-sqrt-5-3-2-frac-2-3-3-frac-1-3\"><pre>\u7b54\u6848:(1)\\frac{\\sqrt{5}}{3},(2)\\frac{2}{3},(3)\\frac{1}{3}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-6670796\" data-v=\"4\" data-block-id=\"6670796\"><style>@media screen and (min-width:690px){.stk-6670796 {flex:var(--stk-flex-grow, 1) 1 calc(29.5\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl1-1.png\" alt=\"\" style=\"width:195px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>2.\u5982\u56fe, \u4e09\u68f1\u9525  A-B C D  \u4e2d,  D A=D B=D C, B D \\perp C D ,  \\angle A D B=\\angle A D C=60^{\\circ}, E  \u4e3a \\\\ B C  \u7684\u4e2d\u70b9.\n(1)\u8bc1\u660e:  B C \\perp D A ;\n(2)\u70b9  F  \u6ee1\u8db3  \\overrightarrow{E F}=\\overrightarrow{D A} , \u6c42\u4e8c\u9762\u89d2  D-A B-F  \u7684\u6b63\u5f26\u503c.<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-c346006\" data-block-id=\"c346006\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-c346006-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-c4aafc0\" data-v=\"4\" data-block-id=\"c4aafc0\"><style>@media screen and (min-width:690px){.stk-c4aafc0 {flex:var(--stk-flex-grow, 1) 1 calc(70\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-1-a-e-perp-b-c-mathrm-bc-perp-d-e-a-e-cap-d-e-e-rightarrow-b-c-perp-ade-rightarrow-b-c-perp-ad-2-d-sqrt-2-0-0-b-0-sqrt-2-0-a-0-0-sqrt-2-f-sqrt-2-0-sqrt-2-rightarrow-d-a-b-boldsymbol-m-1-1-1-rightarrow-a-b-f-boldsymbol-n-0-1-1-cos-langle-boldsymbol-m-boldsymbol-n-rangle-frac-sqrt-6-3-sin-theta-frac-sqrt-3-3\"><pre>\u63d0\u793a:(1)  A E \\perp B C, \\mathrm{BC} \\perp D E, A E \\cap D E=E, ~~~~~~~~~~~~~~~\\\\\\Rightarrow B C \\perp  \u9762 ADE,\\Rightarrow B C \\perp AD.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2)\u5efa\u7cfb\u5982\u56fe\uff0c  D(\\sqrt{2}, 0,0), B(0, \\sqrt{2}, 0), A(0,0, \\sqrt{2}) ,~~~~~~~~~\\\\\n\nF(-\\sqrt{2}, 0, \\sqrt{2}),\\Rightarrow \u5e73\u9762  D A B  \u7684\u6cd5\u5411\u91cf\u4e3a  \\boldsymbol{m}=(1,1,1) .\\Rightarrow \\\\\n\n\u5e73\u9762  A B F  \u7684\u6cd5\u5411\u91cf\u4e3a  \\boldsymbol{n}=(0,1,1) ,\n\u6240\u4ee5  \\cos \\langle\\boldsymbol{m}, \\boldsymbol{n}\\rangle=\\frac{\\sqrt{6}}{3},\\\\\n\n\\sin \\theta=\\frac{\\sqrt{3}}{3}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-078bdec\" data-v=\"4\" data-block-id=\"078bdec\"><style>@media screen and (min-width:690px){.stk-078bdec {flex:var(--stk-flex-grow, 1) 1 calc(30\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"354\" height=\"293\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl4.png\" alt=\"\" class=\"wp-image-3957\" style=\"width:187px;height:auto\" srcset=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl4.png 354w, http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl4-300x248.png 300w\" sizes=\"auto, (max-width: 354px) 100vw, 354px\" \/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>3.\u5df2\u77e5\u4e00\u5706\u5f62\u7eb8\u7247\u7684\u5706\u5fc3\u4e3a  O , \u76f4\u5f84  A B=2 , \u5706\u5468\u4e0a\u6709  C, D  \u4e24\u70b9. \u5982\u56fe,  O C \\perp A B , ~~~~~~~~~~~\\\\ \\angle A O D=\\frac{\\pi}{6} , \u70b9  P  \u662f   \\stackrel\\frown{BD} \u4e0a\u7684\u52a8\u70b9\u6cbf  A B  \u5c06\u7eb8\u7247\u6298\u4e3a\u76f4\u4e8c\u9762\u89d2, \u5e76\u8fde\u63a5  P O, P D ,  P C, C D .\\\\\n(1)\u5f53  A B \/ \/  \u5e73\u9762  P C D  \u65f6, \u6c42  P D  \u7684\u957f;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2)\u5f53\u4e09\u68f1\u9525  P-C O D  \u7684\u4f53\u79ef\u6700\u5927\u65f6, \u6c42\u5e73\u9762  O P D  \u4e0e\u5e73\u9762  C P D  \u5939\u89d2\u7684\u4f59\u5f26\u503c.~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-576bd6d\" data-block-id=\"576bd6d\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-576bd6d-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-8094a5b\" data-v=\"4\" data-block-id=\"8094a5b\"><style>@media screen and (min-width:690px){.stk-8094a5b {flex:var(--stk-flex-grow, 1) 1 calc(50\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u8f85\u52a9\u7ebf<\/summary>\n<p id=\"htoc-111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl5-1.png\" alt=\"\" style=\"width:212px;height:auto\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-9df3850\" data-v=\"4\" data-block-id=\"9df3850\"><style>@media screen and (min-width:690px){.stk-9df3850 {flex:var(--stk-flex-grow, 1) 1 calc(50\n<figure class=\"wp-block-image alignright size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl5.png\" alt=\"\" style=\"width:340px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-1-a-b-p-c-d-rightarrow-a-b-p-d-rightarrow-angle-p-o-d-frac-2-pi-3-rightarrow-p-d-sqrt-3-2-o-c-perp-p-o-d-o-d-perp-o-p-p-c-o-d-c-1-0-0-d-0-0-1-p-0-1-0-overrightarrow-p-c-1-1-0-overrightarrow-d-p-0-1-1-rightarrow-c-p-d-boldsymbol-n-1-1-1-1-rightarrow-o-p-d-boldsymbol-n-2-1-0-0-rightarrow-cos-left-langle-overrightarrow-n-1-overrightarrow-n-2-right-rangle-frac-sqrt-3-3\"><pre>\u63d0\u793a: (1)\u56e0\u4e3a  A B \/ \/  \u5e73\u9762  P C D, \\Rightarrow A B \/ \/ P D, \\Rightarrow \\angle P O D=\\frac{2 \\pi}{3}, \\Rightarrow P D=\\sqrt{3};~~~~~~~~~~~~~~~~~\\\\\n(2) \u7531\u9898\u610f\u77e5  O C \\perp  \u5e73\u9762  P O D , \u4e14  O D \\perp O P  \u65f6, \u4e09\u68f1\u96c9  P-C O D  \u7684\u4f53\u79ef\u6700\u5927.\u5efa\u7cfb\u5982\u56fe~~\\\\\u6240\u793a,\u5219  C(1,0,0), D(0,0,1), P(0,1,0) ,\n\u6545  \\overrightarrow{P C}=(1,-1,0), \\overrightarrow{D P}=(0,1,-1),\n \\Rightarrow~~~~~~~~~~~~~~~  \\\\\u5e73\u9762  C P D  \u7684\u4e00\u4e2a\u6cd5\u5411\u91cf\u4e3a  \\boldsymbol{n}_{1}=(1,1,1),\n \\Rightarrow  \u5e73\u9762  O P D  \u7684\u4e00\u4e2a\u6cd5\u5411\u91cf\u4e3a  \\boldsymbol{n}_{2}=(1,0,0) ,~~~~\\\\\n\\Rightarrow   \\cos \\left\\langle\\overrightarrow{n_{1}}, \\overrightarrow{n_{2}}\\right\\rangle=\\frac{\\sqrt{3}}{3} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>\u7ec3\u4e60.\u5982\u56fe,\u56db\u68f1\u96c9  P-A B C D  \u7684\u6240\u6709\u9876\u70b9\u5747\u5728\u540c\u4e00\u4e2a\u7403\u7684\u7403\u9762\u4e0a,\u4e14  A B=A D=4, ~~\\\\B C \\perp C D, P B \\perp  \u5e73\u9762  P A D  .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(1)\u8bc1\u660e\uff1a\u5e73\u9762  P A B \\perp  \u5e73\u9762  A B C D  ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2)\u6c42\u56db\u68f1\u96c9  P-A B C D  \u4f53\u79ef\u7684\u6700\u5927\u503c;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(3)\u5f53\u56db\u68f1\u96c9  P-A B C D  \u7684\u4f53\u79ef\u6700\u5927\u65f6,\u6c42\u76f4\u7ebf  P C  \u4e0e\u5e73\u9762  P B D  \u6240\u6210\u89d2\u7684\u6b63\u5f26\u503c.~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-28f84493 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:66.66%\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-da\"><pre>\u63d0\u793a:(1)AD\\bot AB,AD\\bot PB.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2)ABCD\u56db\u70b9\u5171\u5706,BD\u4e3a\u76f4\u5f84,AC\\bot BD\u65f6\uff0c~~~~~~\\\\S_{ABCD}\u6700\u5927,\u540c\u65f6PB\\bot PA,AB\u4e3a\u76f4\u5f84,~~~~~~~~~~~~~~~~~~~~\\\\\\therefore PA=PB\u65f6,\u9ad8\u6700\u5927,\u6b64\u65f6V=\\frac{32}{3}.~~~~~~~~~~~~~~~~~~~~~~\\\\\n(3)\\frac{\\sqrt{2}}{3}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:33.33%\">\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl10.png\" alt=\"\"\/><\/figure>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>4.\u5982\u56fe\uff0c\u56db\u68f1\u96c9  P-A B C D  \u4e2d\uff0c  P A \\perp  \u5e95\u9762  A B C D, P A=A C=2, B C=1, A B=\\sqrt{3} .\\\\\n(1) \u82e5  A D \\perp P B  \uff0c\u8bc1\u660e:  A D \/ \/  \u5e73\u9762  P B C ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2) \u82e5  A D \\perp D C  \uff0c\u4e14\u4e8c\u9762\u89d2  A-C P-D  \u7684\u6b63\u5f26\u503c\u4e3a  \\frac{\\sqrt{42}}{7}  \uff0c\u6c42  A D .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-7dc53b5\" data-block-id=\"7dc53b5\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-7dc53b5-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-82c8f46\" data-v=\"4\" data-block-id=\"82c8f46\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-82c8f46-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-82c8f46-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u8f85\u52a9\u7ebf<\/summary>\n<p id=\"htoc-11111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxlzm1-1.png\" alt=\"\" style=\"width:181px;height:auto\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-a62d471\" data-v=\"4\" data-block-id=\"a62d471\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-a62d471-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-a62d471-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u8f85\u52a9\u7ebf<\/summary>\n<p id=\"htoc-111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxlzm1-3.png\" alt=\"\" style=\"width:199px;height:auto\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-630e56e\" data-v=\"4\" data-block-id=\"630e56e\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-630e56e-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-630e56e-inner-blocks\">\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxlzm1-2.png\" alt=\"\" style=\"width:169px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-abcd-de-perp-ac-e-em-perp-pc-dm-angle-dme-a-c-p-d-theta-ec-t-em-frac-sqrt-2-t-2-de-sqrt-t-2-t-tan-theta-frac-de-em-sqrt-6-rightarrow-t-frac-1-2-rightarrow-ad-sqrt-3\"><pre>\u6cd5\u4e00\u3001\u51e0\u4f55\u6cd5.\u5728\u5e95\u9762ABCD\u4e2d\u4f5cDE\\perp AC \u4e8e\u70b9E,\u4f5cEM\\perp PC,\u8fde\u63a5DM,\u5219\\angle DME~ \\\\\u4e3a\u4e8c\u9762\u89d2  A-C P-D  \u7684\u5e73\u9762\u89d2\\theta ,\u8bbeEC=t,\u5219EM=\\frac{\\sqrt{2}t }{2} ,DE=\\sqrt{t(2-t)} ,\\tan \\theta =~~\\\\\\frac{DE}{EM} =\\sqrt{6},\\Rightarrow t=\\frac{1}{2} ,\\Rightarrow AD=\\sqrt{3}.  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-ad-t-a-t-0-0-p-t-0-2-c-0-sqrt-4-t-2-0-rightarrow-apc-overrightarrow-n-1-1-frac-t-sqrt-4-t-2-0-cdp-overrightarrow-n-2-2-0-t-cos-lt-overrightarrow-n-1-overrightarrow-n-2-gt-frac-sqrt-7-7-rightarrow-ad-sqrt-3\"><pre>\u6cd5\u4e8c\u3001\u5750\u6807\u6cd5.\u5efa\u7cfb\u5982\u56fe,\u8bbeAD=t,\u5219A(t,0,0),P(t,0,2),C(0,\\sqrt{4-t^2},0 ),\\Rightarrow \u5e73\u9762APC\\\\\u7684\u6cd5\u5411\u91cf\\overrightarrow{ n_1}=(1,\\frac{t}{\\sqrt{4-t^2} } ,0),\u5e73\u9762CDP\u7684\u6cd5\u5411\u91cf\\overrightarrow{ n_2}=(-2,0,t),\\cos &lt;\\overrightarrow{ n_1},\\overrightarrow{ n_2}&gt;=\\frac{\\sqrt{7} }{7} ,\\\\\\Rightarrow AD=\\sqrt{3} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>5.\u5982\u56fe, \u5728\u4e09\u68f1\u9525  A-B C D  \u4e2d, \u5e73\u9762  A B D \\perp  \u5e73\u9762  B C D ,  A B=A D, O  \u4e3a  B D  \u7684\u4e2d\u70b9.~~~~~~~\\\\\n(1)\u8bc1\u660e:  O A \\perp C D ;\n(2)\u82e5  \\triangle O C D  \u662f\u8fb9\u957f\u4e3a 1 \u7684\u7b49\u8fb9\u4e09\u89d2\u5f62, \u70b9  E  \u5728\u68f1  A D  \u4e0a,  D E=2 E A ,\\\\ \u4e14\u4e8c\u9762\u89d2  E-B C-D  \u7684\u5927\u5c0f\u4e3a  45^{\\circ} , \u6c42\u4e09\u68f1\u9525  A-B C D  \u7684\u4f53\u79ef.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-6922dd4\" data-block-id=\"6922dd4\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-6922dd4-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-46ab36a\" data-v=\"4\" data-block-id=\"46ab36a\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-46ab36a-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-46ab36a-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u8f85\u52a9\u7ebf<\/summary>\n<p id=\"htoc-11111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxlzm2-2.png\" alt=\"\" style=\"width:196px;height:auto\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-4dd68e4\" data-v=\"4\" data-block-id=\"4dd68e4\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-4dd68e4-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-4dd68e4-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u8f85\u52a9\u7ebf<\/summary>\n<p id=\"htoc-111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxlzm2-1.png\" alt=\"\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-367f30b\" data-v=\"4\" data-block-id=\"367f30b\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-367f30b-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-367f30b-inner-blocks\">\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxlzm2.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-e-e-f-a-o-b-d-f-f-f-g-perp-b-c-g-e-g-angle-e-g-f-45-circ-g-f-e-f-d-e-2-e-a-therefore-e-f-frac-2-3-o-a-d-f-2-o-f-therefore-frac-b-f-f-d-2-therefore-g-f-frac-2-3-therefore-e-f-g-f-frac-2-3-therefore-o-a-1-therefore-v-a-b-c-d-frac-1-3-s-triangle-b-c-d-cdot-a-o-frac-sqrt-3-6\"><pre>\u65b9\u6cd5\u4e00\u3001\u5982\u56fe, \u8fc7\u70b9  E  \u4f5c  E F \/ \/ A O , \u4ea4  B D  \u4e8e  F , \u8fc7\u70b9  F  \u4f5c  F G \\perp B C , \u5782\u8db3\u4e3a  G ,\u8fde\u63a5  E G ,~~\\\\ \\angle E G F=45^{\\circ} , \u5219  G F=E F \u7531  D E=2 E A ,\\therefore E F=\\frac{2}{3} O A, D F=2 O F, \n\\therefore \\frac{B F}{F D}=2 .~~~~~~~ \\\\\\therefore G F=\\frac{2}{3}, \\therefore E F=G F=\\frac{2}{3}, \\therefore O A=1, \\therefore V_{A-B C D}=\\frac{1}{3} S_{\\triangle B C D} \\cdot A O =\\frac{\\sqrt{3}}{6} .~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-b-1-0-0-d-1-0-0-c-left-frac-1-2-frac-sqrt-3-2-0-right-a-0-0-a-a-gt-0-e-left-frac-1-3-0-frac-2-a-3-right-b-c-d-boldsymbol-n-0-0-1-b-c-e-boldsymbol-m-left-1-sqrt-3-frac-2-a-right-left-therefore-cos-langle-boldsymbol-m-cdot-boldsymbol-n-gt-frac-boldsymbol-m-cdot-boldsymbol-n-boldsymbol-m-boldsymbol-n-mid-right-rvert-frac-sqrt-2-2-a-1-o-a-1-s-triangle-b-c-d-frac-1-2-b-d-cdot-c-d-sin-60-circ-frac-sqrt-3-2-therefore-v-a-b-c-d-frac-1-3-s-triangle-b-c-d-cdot-o-a-frac-sqrt-3-6\"><pre>\u65b9\u6cd5\u4e8c\u3001\u5efa\u7cfb\u5982\u56fe\uff0c  B(1,0,0), D(-1,0,0), C\\left(-\\frac{1}{2}, \\frac{\\sqrt{3}}{2}, 0\\right) . \u8bbe  A(0  \uff0c  0, a), a&gt;0, ~~~~~~~~~\\\\E\\left(-\\frac{1}{3}, 0, \\frac{2 a}{3}\\right) .\n\n\u5e73\u9762  B C D  \u7684\u4e00\u4e2a\u6cd5\u5411\u91cf\u4e3a  \\boldsymbol{n}=(0,0,1) , \u5e73\u9762  B C E  \u7684\u6cd5\u5411\u91cf\u4e3a~~~~~~~~~~~~~~~ \\\\ \\boldsymbol{m}=\\left(1, \\sqrt{3}, \\frac{2}{a}\\right),  \\left.\\therefore \\cos \\langle\\boldsymbol{m} \\cdot \\boldsymbol{n}&gt;=| \\frac{\\boldsymbol{m} \\cdot \\boldsymbol{n}}{|\\boldsymbol{m}| |\\boldsymbol{n} \\mid} \\right\\rvert\\,=\\frac{\\sqrt{2}}{2} , \u5f97  a=1 , \u5373  O A=1, S_{\\triangle B C D}=~~~~~~~\\\\\\frac{1}{2} B D \\cdot C D \\sin 60^{\\circ}=\\frac{\\sqrt{3}}{2} ,  \\therefore V_{A-B C D}=\\frac{1}{3} S_{\\triangle B C D} \\cdot O A=\\frac{\\sqrt{3}}{6} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>6.\u5982\u56fe, \u5728\u76f4\u4e09\u68f1\u67f1  A B C-A_{1} B_{1} C_{1}  \u4e2d,  \\angle B A C=90^{\\circ}, A B=A C=2 ,  A A_{1}=3 . M  \u662f  A B\\\\  \u7684\u4e2d\u70b9,  N  \u662f  B_{1} C_{1}  \u7684\u4e2d\u70b9,  P  \u662f  B C_{1}  \u4e0e  B_{1} C \u7684\u4ea4\u70b9. \u5728\u7ebf\u6bb5  A_{1} N  \u4e0a\u662f\u5426\u5b58\u5728\u70b9  Q , \u4f7f\u5f97 ~~~~~~~ \\\\P Q \/ \/  \u5e73\u9762  A_{1} C M  ?~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-a5c829a\" data-block-id=\"a5c829a\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-a5c829a-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-3c7a306\" data-v=\"4\" data-block-id=\"3c7a306\"><style>@media screen and (min-width:690px){.stk-3c7a306 {flex:var(--stk-flex-grow, 1) 1 66.7\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-q-q-n-overrightarrow-a-1q-lambda-overrightarrow-a-1n-overrightarrow-pq-overrightarrow-pa-1-overrightarrow-a-1q\"><pre>\u7b54\u6848:\u5b58\u5728Q\uff0cQ\u4e3a\u9760\u8fd1N\u7684\u4e09\u7b49\u5206\u70b9;~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\u63d0\u793a:\u8bbe\\overrightarrow{A_1Q}=\\lambda \\overrightarrow{A_1N},\\overrightarrow{PQ}=\\overrightarrow{PA_1}+\\overrightarrow{A_1Q}.~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-82adc6d\" data-v=\"4\" data-block-id=\"82adc6d\"><style>@media screen and (min-width:690px){.stk-82adc6d {flex:var(--stk-flex-grow, 1) 1 33.3\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl6.png\" alt=\"\" style=\"width:180px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>7.\u5982\u56fe,\u5df2\u77e5\u7b49\u8170\u68af\u5f62  A B C D, A B=4, C D=6, A D=\\sqrt{5}, E, F  \u5206\u522b\u4e3a  A B, C D  \u7684\u4e2d~~~~\\\\\u70b9,\u6cbf\u7ebf\u6bb5  E F  \u5c06\u56db\u8fb9\u5f62  A E F D  \u7ffb\u6298\u5230\u56db\u8fb9\u5f62  E F N M  \u7684\u4f4d\u7f6e,\u70b9  P  \u4e3a\u7ebf\u6bb5  N C  \u4e0a\u4e00\u70b9,~~~~~~~\\\\\u4e14\u6ee1\u8db3  \\overrightarrow{N P}=\\frac{2}{3} \\overrightarrow{N C} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(1)\u8bc1\u660e: B P \/ \/  \u5e73\u9762  E F M ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2)\u8bbe\u4e8c\u9762\u89d2  M-E F-B  \u7684\u5e73\u9762\u89d2\u4e3a  \\theta(0&lt;\\theta&lt;\\pi),\u5728\u56db\u8fb9\u5f62  A E F D  \u7ffb\u6298\u8fc7\u7a0b\u4e2d,\u662f~~~~~~\\\\\u5426\u5b58\u5728  \\theta ,\u4f7f\u5f97  M F \u4e0e\u5e73\u9762  E B P  \u6240\u6210\u89d2\u7684\u6b63\u5f26\u503c\u4e3a  \\frac{3 \\sqrt{10}}{10} ,\u82e5\u5b58\u5728,\u8bf7\u8bf4\u660e\u7406\u7531\uff0e~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-ba3fc6c\" data-block-id=\"ba3fc6c\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-ba3fc6c-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-74abdd4\" data-v=\"4\" data-block-id=\"74abdd4\"><style>@media screen and (min-width:690px){.stk-74abdd4 {flex:var(--stk-flex-grow, 1) 1 66.7\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111\"><\/p>\n\n\n\n<p id=\"htoc-11111111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2024\/11\/ltjhjx7-2-1-1-2.png\" alt=\"\" class=\"wp-image-4129\" style=\"width:256px;height:auto\"\/><\/figure>\n\n\n\n<p id=\"htoc-111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-d\"><pre>\u7b54\u6848:\\theta=90^{\\circ};~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-t\"><pre>\u63d0\u793a:FN=3,\\angle CFN=\\theta ,\\Rightarrow N(3\\cos\\theta ,3\\sin\\theta ,0),~~~~\\\\M(2\\cos\\theta ,2\\sin\\theta ,2),\\overrightarrow{CP} =\\lambda \\overrightarrow{CN}\\Rightarrow ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\P(3\\lambda \\cos\\theta -3\\lambda +3,3\\sin\\theta ,0).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-0258e26\" data-v=\"4\" data-block-id=\"0258e26\"><style>@media screen and (min-width:690px){.stk-0258e26 {flex:var(--stk-flex-grow, 1) 1 33.3\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/ltjhjx7-2-1-1.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>8.\u5728  \\triangle N B C  \u4e2d, \\angle B=90^{\\circ}, A D \/ \/ B C, N A=C D=2 A B=2  ,\u5982\u56fe\u5c06  \\triangle N A D  \u6cbf  A D  \u7ffb~\\\\\u6298\u81f3  \\triangle P A D  .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(1)\u8bc1\u660e\uff1a\u5e73\u9762  P B C \\perp  \u5e73\u9762  P A B  ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2)\u82e5\u4e8c\u9762\u89d2  P-A D-B  \u5927\u5c0f\u4e3a  120^{\\circ} ,\n\u5728\u7ebf\u6bb5  P D  \u4e0a\u662f\u5426\u5b58\u5728\u70b9  E  ,\u4f7f\u5f97\u5e73\u9762  A B E  \u4e0e~~~~\\\\\u5e73\u9762  P D C  \u6240\u6210\u89d2\u7684\u4f59\u5f26\u503c\u4e3a  \\frac{1}{5}  ?\u82e5\u5b58\u5728,\u786e\u5b9a\u70b9  E \u7684\u4f4d\u7f6e;\u82e5\u4e0d\u5b58\u5728,\u8bf4\u660e\u7406\u7531.~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-5d80f8a\" data-block-id=\"5d80f8a\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-5d80f8a-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-4d93e95\" data-v=\"4\" data-block-id=\"4d93e95\"><style>@media screen and (min-width:690px){.stk-4d93e95 {flex:var(--stk-flex-grow, 1) 1 calc(61.29999999999998\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u8f85\u52a9\u7ebf<\/summary>\n<p id=\"htoc-1111111111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl8-1.png\" alt=\"\" style=\"width:265px;height:auto\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-88ab5da\" data-v=\"4\" data-block-id=\"88ab5da\"><style>@media screen and (min-width:690px){.stk-88ab5da {flex:var(--stk-flex-grow, 1) 1 calc(38.70000000000002\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl8.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-t1\"><pre>\u63d0\u793a:(1)\u8bc1\u660eAD\\perp\u9762PAB;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2) A(0,0,0), P(-1,0, \\sqrt{3}), B(1,0,0), D(0,2 \\sqrt{3}, 0), C(1,3 \\sqrt{3}, 0)  ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n \\overrightarrow{D P}=(-1,-2 \\sqrt{3}, \\sqrt{3})  ,\u8bbe  \\overrightarrow{D E}=\\lambda \\overrightarrow{D P}, \\lambda \\in(0,1)  ,\\boxed{ \\overrightarrow{AE}= \\overrightarrow{AD}  +\\overrightarrow{D E}},\\Rightarrow ~~~~~~~~~~~~~~~~~~~~~\n\\\\\\overrightarrow{A E}=(-\\lambda, 2 \\sqrt{3}(1-\\lambda), \\sqrt{3} \\lambda), \\overrightarrow{A B}=   (1,0,0), \\overrightarrow{D C}=(1, \\sqrt{3}, 0)  ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\u8bbe\u5e73\u9762  A B E  \u7684\u6cd5\u5411\u91cf\u4e3a  \\vec{m}=(x, y, z)  ,\u5e73\u9762  P D C \u7684\u6cd5\u5411\u91cf\u4e3a  \\vec{n}=(a, b, c)  ,~~~~~~~~~~~~~~~~~~~~~\\\\\n \\left\\{\\begin{array}{l}\\vec{m} \\cdot \\overrightarrow{A B}=0 \\\\ \\vec{m} \\cdot \\overrightarrow{A E}=0\\end{array}\\right.  ,\u5373  \\left\\{\\begin{array}{c}x=0 \\\\ -\\lambda x+2 \\sqrt{3}(1-\\lambda) y+\\sqrt{3} \\lambda z=0\\end{array}\\right.  ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\\boxed{\u4ee4  y=\\lambda ,\n\u53ef\u5f97  \\vec{m}=(0, \\lambda, 2(\\lambda-1))},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\u540c\u7406\u53ef\u5f97  \\vec{n}=(\\sqrt{3},-1,-1)  ,\u8bbe\u5e73\u9762  A B E  \u4e0e\u5e73\u9762  P D C  \u7684\u5939\u89d2\u4e3a  \\theta  ,  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\\\\\cos \\theta=\\frac{|3 \\lambda-2|}{\\sqrt{5} \\cdot \\sqrt{5 \\lambda^{2}-8 \\lambda+4}}=\\frac{1}{5}  ,\u89e3\u5f97  \\lambda=\\frac{1}{2}  \u6216  \\frac{4}{5}  ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\u6240\u4ee5\u5b58\u5728\u70b9  E  ,\u5f53  D E=\\frac{1}{2} D P \u6216  D E=\\frac{4}{5} D P  \u65f6\u6ee1\u8db3\u9898\u610f.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>9.\u5982\u56fe,\u5728\u4e09\u68f1\u67f1  A B C-A_{1} B_{1} C_{1}  \u4e2d,\u5e95\u9762  A B C  \u662f\u7b49\u8170\u76f4\u89d2\u4e09\u89d2\u5f62,\u4e14  A B=A C, ~~~~~~~~\\\\D, E  \u5206\u522b\u662f  B C, B_{1} C_{1}  \u7684\u4e2d\u70b9,\u4e14  A_{1} D \\perp  \u5e73\u9762  A B C  .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(1)\u8bc1\u660e:\u4fa7\u9762  B C C_{1} B_{1}  \u4e3a\u77e9\u5f62;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2)\u82e5  A A_{1}=2 A B=4 ,\u6c42\u5e73\u9762  A_{1} C E  \u4e0e\u5e73\u9762  A C C_{1} A_{1}  \u7684\u5939\u89d2\u7684\u4f59\u5f26\u503c.~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-126d92b\" data-block-id=\"126d92b\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-126d92b-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-df2021b\" data-v=\"4\" data-block-id=\"df2021b\"><style>@media screen and (min-width:690px){.stk-df2021b {flex:var(--stk-flex-grow, 1) 1 calc(63.29999999999998\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u8f85\u52a9\u7ebf<\/summary>\n<p id=\"htoc-11111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl9-2.png\" alt=\"\" style=\"width:270px;height:auto\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-4422ccd\" data-v=\"4\" data-block-id=\"4422ccd\"><style>@media screen and (min-width:690px){.stk-4422ccd {flex:var(--stk-flex-grow, 1) 1 calc(36.70000000000002\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl9-1.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-ad-d-db-ad-da-x-y-z-a-1-e-ad-sqrt-2-a-1-d-sqrt-14-therefore-a-0-sqrt-2-0-c-sqrt-2-0-0-a-1-0-0-sqrt-14-e-sqrt-2-0-sqrt-2-overrightarrow-ca-1-sqrt-2-0-sqrt-14-overrightarrow-ce-sqrt-2-sqrt-2-sqrt-14-overrightarrow-aa-1-0-sqrt-2-sqrt-14\"><pre>\u2605\u60ac\u7a7a\u70b9\u7684\u5750\u6807\u2605\\\\\n(2)\u7b54\u6848:\u5e73\u9762A_{1}CE\u4e0e\u5e73\u9762ACC_{1}A_{1}\u7684\u5939\u89d2\u7684\u4f59\u5f26\u503c\u4e3a\\frac{2\\sqrt{30}}{15}. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\\\\u63d0\u793a:A_{1}E = AD=\\sqrt{2},A_{1}D=\\sqrt{14},\\therefore A(0,-\\sqrt{2},0),C(-\\sqrt{2},0,0),A_{1}(0,0,\\sqrt{14}),~~\\\\\nE(\\sqrt{2},0,\\sqrt{2})\n\u5219\\overrightarrow{CA_{1}}=(\\sqrt{2},0,\\sqrt{14}),\\overrightarrow{CE}=(\\sqrt{2},\\sqrt{2},\\sqrt{14}),\\overrightarrow{AA_{1}}=(0,\\sqrt{2},\\sqrt{14}).~~~~~~~~\\\\\n\u8bbe\u5e73\u9762A_{1}CE\u7684\u6cd5\u5411\u91cf\u4e3a\\boldsymbol{n}=(x,y,z),\u6613\u5f97\n\\boldsymbol{n}=(\\sqrt{7},0,-1),~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\\\\u8bbe\u5e73\u9762ACC_{1}A_{1}\u7684\u6cd5\u5411\u91cf\u4e3a\\boldsymbol{m}=(a,b,c),\u6613\u5f97\\boldsymbol{m}=(\\sqrt{7},\\sqrt{7},-1).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\\therefore|\\cos\\langle\\boldsymbol{m},\\boldsymbol{n}\\rangle|=\\frac{|\\boldsymbol{m}\\cdot\\boldsymbol{n}|}{|\\boldsymbol{m}|\\cdot|\\boldsymbol{n}|}=\\frac{8}{\\sqrt{8}\\times\\sqrt{15}}=\\frac{2\\sqrt{30}}{15},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>10.\u5982\u56fe,\u70b9D\u662f\u4ee5AB\u4e3a\u76f4\u5f84\u7684\u534a\u5706\u4e0a\u7684\u52a8\u70b9,\u5df2\u77e5AB = BC = 3,\u4e14AB\\perp BC,\u5e73~~~~~\\\\\u9762BCD\\perp\u5e73\u9762ACD.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n \n(1)\u8bc1\u660e:BD\\perp BC;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2)\u82e5\u7ebf\u6bb5AC\u4e0a\u5b58\u5728\u4e00\u70b9E\u6ee1\u8db3\\overrightarrow{CE}=2\\overrightarrow{EA},\u5f53\u4e09\u68f1\u9525C - ABD\u7684\u4f53\u79ef\u53d6\u5f97\u6700\u5927\u503c~~~~\\\\\u65f6,\u6c42\u5e73\u9762BED\u4e0e\u5e73\u9762AEB\u5939\u89d2\u7684\u4f59\u5f26\u503c.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-99c8684\" data-block-id=\"99c8684\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-99c8684-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-24cf9ff\" data-v=\"4\" data-block-id=\"24cf9ff\"><style>@media screen and (min-width:690px){.stk-24cf9ff {flex:var(--stk-flex-grow, 1) 1 calc(30.167\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-d44d5df\" data-v=\"4\" data-block-id=\"d44d5df\"><style>@media screen and (min-width:690px){.stk-d44d5df {flex:var(--stk-flex-grow, 1) 1 calc(36.5\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u56fe\u8c61<\/summary>\n<p id=\"htoc-111111111111111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl10-2.png\" alt=\"\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-2f854ef\" data-v=\"4\" data-block-id=\"2f854ef\"><style>@media screen and (min-width:690px){.stk-2f854ef {flex:var(--stk-flex-grow, 1) 1 calc(33.333\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl10-1.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-ti\"><pre>\u7b54\u6848(2)\\frac{\\sqrt{6}}{6};~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\u63d0\u793a:(1)\u4f5cBH\\perp CD ,\\therefore  BH\\perp \u5e73\u9762ACD.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2)\u89e3:\u636e(1)\u77e5,\u5f53\u4e14\u4ec5\u5f53BD = AD\u65f6,V_{C - ABD}\u8fbe\u5230\u6700\u5927,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\u8bbe\u5e73\u9762BED\u7684\u6cd5\u5411\u91cf\u4e3a\\overrightarrow{n_{1}}=(x,y,z),~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\u5219\u70b9B(0,-\\frac{3}{2},0),A(0,\\frac{3}{2},0),D(\\frac{3}{2},0,0),E(0,\\frac{1}{2},1),C(0,-\\frac{3}{2},3),\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\\therefore\\overrightarrow{BD}=(\\frac{3}{2},\\frac{3}{2},0),\n\\overrightarrow{BE}=(0,2,1),\\Rightarrow\\begin{cases}x + y = 0\\\\2y + z = 0\\end{cases},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\u6cd5\u5411\u91cf\u4e3a\\overrightarrow{n_{1}}=(1,-1,2),\u56e0\u4e3a\u5e73\u9762AEB\u7684\u4e00\u4e2a\u6cd5\u5411\u91cf\u4e3a\\overrightarrow{n_{2}}=(1,0,0),~~~~~~~~~~~~~~~~~~~~~~~~\\\\\\vert\\cos\\langle\\overrightarrow{n_{1}},\\overrightarrow{n_{2}}\\rangle\\vert=\\frac{\\vert\\overrightarrow{n_{1}}\\cdot\\overrightarrow{n_{2}}\\vert}{\\vert\\overrightarrow{n_{1}}\\vert\\cdot\\vert\\overrightarrow{n_{2}}\\vert}=\\frac{1}{\\sqrt{4 + 1+1}\\times1}=\\frac{\\sqrt{6}}{6}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>11.\u5982\u56fe,\u5728\u56db\u68f1\u96c9  P-A B C D  \u4e2d,\u5e95\u9762  A B C D  \u662f\u68af\u5f62, A B \/ \/ C D, A B=2 C D=2, ~~~~~~~\\\\A D=4, \\angle B A D=   60^{\\circ}, P D \\perp C D, E  \u4e3a  A B  \u7684\u4e2d\u70b9, M  \u4e3a  C E  \u7684\u4e2d\u70b9\uff0e~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(1)\u8bc1\u660e: P M \\perp A B  ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2)\u82e5  P A=\\sqrt{15}, N  \u4e3a  P C  \u4e2d\u70b9,\u4e14  A N  \u4e0e\u5e73\u9762  P D M  \u6240\u6210\u89d2\u7684\u6b63\u5f26\u503c\u4e3a  \\frac{\\sqrt{15}}{6}  ,\u6c42\u56db\u68f1\u96c9 ~\\\\ P-A B C D \u7684\u4f53\u79ef\uff0e~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-e710f21\" data-block-id=\"e710f21\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-e710f21-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-bd5d666\" data-v=\"4\" data-block-id=\"bd5d666\"><style>@media screen and (min-width:690px){.stk-bd5d666 {flex:var(--stk-flex-grow, 1) 1 calc(29.2\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-d144921\" data-v=\"4\" data-block-id=\"d144921\"><style>@media screen and (min-width:690px){.stk-d144921 {flex:var(--stk-flex-grow, 1) 1 calc(37.46700000000001\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u56fe\u8c61<\/summary>\n<p id=\"htoc-1111111111111111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl11-1.png\" alt=\"\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-2037487\" data-v=\"4\" data-block-id=\"2037487\"><style>@media screen and (min-width:690px){.stk-2037487 {flex:var(--stk-flex-grow, 1) 1 calc(33.333\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl11.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-1-c-d-perp-p-d-m-d-d-b-d-c-x-y-a-b-c-d-z-a-2-sqrt-3-2-0-c-0-1-0-p-x-0-z-n-left-frac-x-2-frac-1-2-frac-z-2-right-overrightarrow-a-n-left-frac-x-2-2-sqrt-3-frac-5-2-frac-z-2-right-p-d-m-vec-n-0-1-0-a-n-p-d-m-theta-sin-theta-cos-frac-vec-n-cdot-overrightarrow-a-n-overrightarrow-a-n-vec-n-frac-frac-5-2-sqrt-left-frac-x-2-2-sqrt-3-right-2-frac-25-4-left-frac-2-2-right-2-frac-sqrt-15-6-x-4-sqrt-3-2-z-2-35-p-a-sqrt-15-x-2-sqrt-3-2-z-2-11-x-sqrt-3-z-2-sqrt-2-v-frac-1-3-s-a-b-c-d-h-frac-1-3-times-3-sqrt-3-times-2-sqrt-2-2\"><pre>\u7b54\u6848:(1)\u7565;(2)V=2 \\sqrt{6};~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\u63d0\u793a:(1)AB\\perp BD,AB\\perp PD.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\(2)\n\u7531(1)\u77e5, C D \\perp  \u5e73\u9762  P D M  ,\u5efa\u7acb\u7a7a\u95f4\u76f4\u89d2\u5750\u6807\u7cfb,\u5219  A(2 \\sqrt{3},-2,0), C(0,1,0)  ,~~~~~~~~\n\\\\\u8bbe \\boxed{ P(x, 0, z) } ,\u5219  N\\left(\\frac{x}{2}, \\frac{1}{2}, \\frac{z}{2}\\right), \\overrightarrow{A N}=\\left(\\frac{x}{2}-2 \\sqrt{3}, \\frac{5}{2}, \\frac{z}{2}\\right)  ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\\\\u5e73\u9762  P D M  \u7684\u4e00\u4e2a\u6cd5\u5411\u91cf\u4e3a  \\vec{n}=(0,1,0)  ,\u8bbe\u76f4\u7ebf  A N  \u4e0e\u5e73\u9762  P D M  \u6240\u6210\u89d2\u4e3a  \\theta  ,~~~~~~~~~~~~~~~~\n\\\\\u5219  \\sin \\theta=|\\cos &lt;\\vec{n}, \\overrightarrow{A N}&gt;|=\\frac{|\\vec{n} \\cdot \\overrightarrow{A N}|}{|\\overrightarrow{A N}||\\vec{n}|}=\\frac{\\sqrt{15}}{6}  ,\u5316\u7b80\u5f97  (x-4 \\sqrt{3})^{2}+z^{2}=35  \uff0e~~~~~~~~~\n\\\\\u7531  |P A|=\\sqrt{15}  ,\u53ef\u5f97  (x-2 \\sqrt{3})^{2}+   z^{2}=11  ,\u6c42\u5f97  x=\\sqrt{3}, z=2 \\sqrt{2}  ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\u6545  V=\\frac{1}{3} S_{A B C D} h=\\frac{1}{3} \\times 3 \\sqrt{3} \\times 2 \\sqrt{2}=2 \\sqrt{6}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>12.\u5982\u56fe,\u5728\u4e09\u68f1\u67f1  A B C-A_{1} B_{1} C_{1}  \u4e2d, \\angle A C B=90^{\\circ}, A_{1} C \\perp A B, A C=1, A A_{1}=2 ,~~~~\\\\  \\angle A_{1} A C=60^{\\circ}  \uff0e~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(1)\u6c42\u8bc1:\u5e73\u9762  A C C_{1} A_{1} \\perp  \u5e73\u9762  A B C  ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2)\u82e5\u76f4\u7ebf  B A_{1}  \u4e0e\u5e73\u9762  B C C_{1} B_{1}  \u6240\u6210\u89d2\u7684\u6b63\u5f26\u503c\u4e3a  \\frac{\\sqrt{2}}{4}  ,\u6c42\u5e73\u9762  A_{1} A B B_{1}  \u4e0e\u5e73\u9762  B C C_{1} B_{1}  \\\\\u6240\u6210\u89d2\u7684\u4f59\u5f26\u503c\uff0e~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-ebbdea0\" data-block-id=\"ebbdea0\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-ebbdea0-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-9950859\" data-v=\"4\" data-block-id=\"9950859\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-9950859-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-9950859-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u56fe\u8c61<\/summary>\n<p id=\"htoc-1111111111111111111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"286\" height=\"347\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl10.png\" alt=\"\" class=\"wp-image-3825\" srcset=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl10.png 286w, http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl10-247x300.png 247w\" sizes=\"auto, (max-width: 286px) 100vw, 286px\" \/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-84c4353\" data-v=\"4\" data-block-id=\"84c4353\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-84c4353-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-84c4353-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u56fe\u8c61<\/summary>\n<p id=\"htoc-111111111111111111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl10-1-1.png\" alt=\"\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-b0d0456\" data-v=\"4\" data-block-id=\"b0d0456\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-b0d0456-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-b0d0456-inner-blocks\">\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl10-2-1.png\" alt=\"\" style=\"width:174px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-1-a-1-c-perp-ac-a-1-c-perp-ab-2-a-1-e-perp-cc-1-a-1-e-perp-bc-cc-1-cap-bc-c-therefore-a-1-e-perp-bc-1-b-1-therefore-sin-angle-a-1-be-frac-sqrt-2-4-a-1-e-frac-sqrt-3-2-rightarrow-ce-frac-3-2-be-frac-sqrt-21-2-rightarrow-bc-sqrt-6-end-align\"><pre>(1) A_{1}C\\perp AC\uff0cA_{1}C\\perp AB.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2) \u6cd5\u4e00\uff1a\u5982\u56fe1\uff0c\u4f5cA_{1}E\\perp CC_{1}\uff0c\u53c8A_{1}E\\perp BC\uff0cCC_{1}\\cap BC = C,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\\therefore A_{1}E\\perp\u5e73\u9762BCC_{1}B_{1},\\therefore\\sin\\angle A_{1}BE=\\frac{\\sqrt{2}}{4},\n \n\u53c8A_{1}E = \\frac{\\sqrt{3}}{2}\\Rightarrow CE=\\frac{3}{2}\uff0c~~~~~~~~~~~~~~~~\\\\BE=\\frac{\\sqrt{21}}{2}\\Rightarrow BC=\\sqrt{3}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\u5efa\u7acb\u5750\u6807\u7cfb\u5982\u56fe\uff0c\u5e73\u9762A_{1}ABB_{1}\u7684\u4e00\u4e2a\u6cd5\u5411\u91cf\u4e3a\\overrightarrow{n_{1}} = (\\sqrt{3},1,1)\uff0c~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\u5e73\u9762BCC_{1}B_{1}\u7684\u4e00\u4e2a\u6cd5\u5411\u91cf\u4e3a\\overrightarrow{n_{2}} = (\\sqrt{3},0,1)~~~~~\n \\\\\n\\therefore\\cos\\langle\\overrightarrow{n_{1}},\\overrightarrow{n_{2}}\\rangle=\\frac{2\\sqrt{5}}{5} \uff0c\\therefore\u5e73\u9762A_{1}ABB_{1}\u4e0e\u5e73\u9762BCC_{1}B_{1}\u6240\u6210\u89d2\u4f59\u5f26\u503c\\frac{2\\sqrt{5}}{5}\n\n~~~~~~~~~~~~~~~~~\\\\<\/pre><\/div>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-c-ca-cb-ca-1-x-y-z-bc-k-k-0-a-1c-sqrt-3-a-1-0-0-sqrt-3-b-0-k-0-c-0-0-0-c-1-1-0-sqrt-3-overrightarrow-ba-1-0-k-sqrt-3-quad-overrightarrow-cb-0-k-0-quad-overrightarrow-cc-1-1-0-sqrt-3-bcc-1b-1-overrightarrow-n-x-y-z-begin-cases-overrightarrow-n-cdot-overrightarrow-cb-0-overrightarrow-n-cdot-overrightarrow-cc-1-0-end-cases-implies-begin-cases-ky-0-x-sqrt-3-z-0-end-cases-x-sqrt-3-y-0-z-1-overrightarrow-n-sqrt-3-0-1-ba-1-bcc-1b-1-theta-sin-theta-frac-overrightarrow-ba-1-cdot-overrightarrow-n-overrightarrow-ba-1-cdot-overrightarrow-n-frac-sqrt-3-sqrt-k-2-3-cdot-sqrt-3-1-frac-sqrt-2-4-k-sqrt-3-overrightarrow-ba-1-0-sqrt-3-sqrt-3-a-b-c-a-1-b-1-c-1-x-ca-y-cb-z-ca-1\"><pre>\u6cd5\u4e8c\u3001\u5efa\u7cfb\u5982\u56fe\uff0c\n\u8bbeBC = k(k &gt; 0)\uff0c\u7531(1)\u77e5A_{1}C=\\sqrt{3}\uff0c\u5219A_{1}(0,0,\\sqrt{3})\uff0c~~~~~~~~~~~~~~~\\\\B(0,k,0)\uff0cC(0,0,0)\uff0cC_{1}(-1,0,\\sqrt{3})\uff0c\u5219\n\\overrightarrow{BA_{1}}=(0,-k,\\sqrt{3})\uff0c\\overrightarrow{CB}=(0,k,0)\uff0c~~~~~~~~~\\\\\\overrightarrow{CC_{1}}=(-1,0,\\sqrt{3})\uff0c\n\u8bbe\u5e73\u9762BCC_{1}B_{1}\u7684\u4e00\u4e2a\u6cd5\u5411\u91cf\\overrightarrow{n}=(x,y,z)\uff0c~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\\\\u53ef\u5f97\\begin{cases}ky = 0\\\\-x+\\sqrt{3}z = 0\\end{cases}\uff0c\n\u5e73\u9762BCC_{1}B_{1}\u7684\u4e00\u4e2a\u6cd5\u5411\u91cf\\overrightarrow{n}=(\\sqrt{3},0,1)\uff0c~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\u8bbe\u76f4\u7ebfBA_{1}\u4e0e\u5e73\u9762BCC_{1}B_{1}\u6240\u6210\u89d2\u4e3a\\theta\uff0c\n\u5219\\sin\\theta=\\frac{\\vert\\overrightarrow{BA_{1}}\\cdot\\overrightarrow{n}\\vert}{\\vert\\overrightarrow{BA_{1}}\\vert\\cdot\\vert\\overrightarrow{n}\\vert}=\\frac{\\sqrt{2}}{4}\uff0c~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\u89e3\u5f97k = \\sqrt{3},\u540c\u4e0a.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n  \n<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>13.\u5982\u56fe,\u56db\u68f1\u96c9  V-A B C D  \u7684\u5e95\u9762  A B C D  \u4e3a\u77e9\u5f62, B C=2 A B=8, V A=V D=~~~~~~~~~\\\\2 \\sqrt{10}, V B=V C=4 \\sqrt{2}  .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(1)\u8bbe\u5e73\u9762  V A D  \u4e0e\u5e73\u9762  V B C  \u7684\u4ea4\u7ebf\u4e3a  l  ,\u8bc1\u660e\uff1a l \/ \/  \u5e73\u9762  A B C D  ;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2)\u82e5\u70b9  M  \u6ee1\u8db3  \\overrightarrow{V M}=\\frac{2}{3} \\overrightarrow{V C}+\\frac{1}{3} \\overrightarrow{C B}  ,\u6c42  M B  \u4e0e\u5e73\u9762  V A D  \u6240\u6210\u89d2\u7684\u6b63\u5f26\u503c.~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-7d0d87f\" data-block-id=\"7d0d87f\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-7d0d87f-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-55cc6ca\" data-v=\"4\" data-block-id=\"55cc6ca\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-55cc6ca-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-55cc6ca-inner-blocks\"><\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-04450ca\" data-v=\"4\" data-block-id=\"04450ca\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-04450ca-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-04450ca-inner-blocks\">\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u53c2\u8003\u56fe\u8c61<\/summary>\n<p id=\"htoc-111111111111111111111111111111111\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl13-2.png\" alt=\"\"\/><\/figure>\n<\/details>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-e5c7016\" data-v=\"4\" data-block-id=\"e5c7016\"><div class=\"stk-column-wrapper stk-block-column__content stk-container stk-e5c7016-container stk--no-background stk--no-padding\"><div class=\"stk-block-content stk-inner-blocks stk-e5c7016-inner-blocks\">\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/09\/kjxl13-1.png\" alt=\"\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-1-2-frac-1-4-1-a-b-c-d-a-d-b-c-a-d-not-subset-v-b-c-b-c-subset-v-b-c-a-d-v-b-c-v-a-d-cap-v-b-c-l-a-d-subset-v-a-d-a-d-l-l-not-subset-a-b-c-d-a-d-subset-a-b-c-d-l-a-b-c-d-2-v-0-0-sqrt-15-a-4-3-0-b-4-1-0-c-4-1-0-d-4-3-0-m-left-x-1-y-1-z-1-right-overrightarrow-v-m-frac-2-3-overrightarrow-v-c-frac-1-3-overrightarrow-c-b-left-x-1-y-1-z-1-sqrt-15-right-frac-2-3-4-1-sqrt-15-frac-1-3-8-0-0-m-left-0-frac-2-3-frac-sqrt-15-3-right-m-triangle-mathrm-vbc-overrightarrow-d-a-8-0-0-overrightarrow-v-a-4-3-sqrt-15-v-a-d-vec-n-left-x-2-y-2-z-2-right-left-begin-array-l-vec-n-cdot-overrightarrow-d-a-8-x-2-0-vec-n-cdot-overrightarrow-v-a-4-x-2-3-y-2-sqrt-15-z-2-0-end-array-right-z-2-3-vec-n-0-sqrt-15-3-overrightarrow-b-m-left-4-frac-1-3-frac-sqrt-15-3-right-cos-langle-overrightarrow-b-m-vec-n-rangle-frac-overrightarrow-b-m-cdot-vec-n-overrightarrow-b-m-cdot-vec-n-frac-1-4-b-m-v-a-d-frac-1-4\"><pre>\u7b54\u6848:(1)\u6027\u8d28\u5b9a\u7406\u8bc1\u660e;(2)\\frac{1}{4};~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\u63d0\u793a:(1)\u7531  A B C D  \u4e3a\u77e9\u5f62,\u5f97  A D \/ \/ B C  ,\u800c  A D \\not \\subset  \u5e73\u9762  V B C  ,\n B C \\subset  \u5e73\u9762  V B C  ,~~~~~~~~~~~~\\\\\u5219  A D \/ \/  \u5e73\u9762  V B C  ,\n\u53c8\u5e73\u9762  V A D \\cap  \u5e73\u9762  V B C=l, A D \\subset  \u5e73\u9762  V A D  ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\u5219  A D \/ \/ l  ,\u53c8  l \\not \\subset  \u5e73\u9762  A B C D, A D \\subset  \u5e73\u9762  A B C D  ,\n\u6240\u4ee5  l \/ \/  \u5e73\u9762  A B C D  .~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2)\u5148\u8bf4\u660e\u5efa\u7cfb\u7684\u6761\u4ef6,\u5efa\u7cfb\u5982\u56fe,\n V(0,0, \\sqrt{15}), A(4,-3,0), B(4,1,0), C(-4,1,0), ~~~~~~~~\\\\D(-4,-3,0)  ,\u8bbe  M\\left(x_{1}, y_{1}, z_{1}\\right)  ,\n\u7531  \\overrightarrow{V M}=\\frac{2}{3} \\overrightarrow{V C}+\\frac{1}{3} \\overrightarrow{C B}  ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\u5f97  \\left(x_{1}, y_{1}, z_{1}-\\sqrt{15}\\right)=\\frac{2}{3}(-4,1,-\\sqrt{15})+\\frac{1}{3}(8,0,0)  ,\n\u5f97  M\\left(0, \\frac{2}{3}, \\frac{\\sqrt{15}}{3}\\right)  ,~~~~~~~~~~~~~~~~~~~\\\\\n\u2605\u6216\u8005\u4e4b\u95f4\u5229\u7528 M \u4e3a  \\triangle \\mathrm{VBC}  \u7684\u91cd\u5fc3,\u76f4\u63a5\u5199\u5750\u6807.\u2605\\\\\n\n\\overrightarrow{D A}=(8,0,0), \\overrightarrow{V A}=(4,-3,-\\sqrt{15})\n\n\u8bbe\u5e73\u9762  V A D  \u7684\u6cd5\u5411\u91cf\u4e3a  \\vec{n}=\\left(x_{2}, y_{2}, z_{2}\\right)  ,~~~~~~~~~\\\\\u5219  \\left\\{\\begin{array}{l}\\vec{n} \\cdot \\overrightarrow{D A}=8 x_{2}=0 ~\\\\ \\vec{n} \\cdot \\overrightarrow{V A}=4 x_{2}-3 y_{2}-\\sqrt{15} z_{2}=0\\end{array}\\right.  ,\n\u53d6  z_{2}=-3  ,\u5f97  \\vec{n}=(0, \\sqrt{15},-3)  ,~~~~~~~~~~~~~~~~~~~~~\\\\\n\u53c8  \\overrightarrow{B M}=\\left(-4,-\\frac{1}{3}, \\frac{\\sqrt{15}}{3}\\right)  ,\u5f97\u5230\n\n|\\cos \\langle\\overrightarrow{B M}, \\vec{n}\\rangle|=\\frac{|\\overrightarrow{B M} \\cdot \\vec{n}|}{|\\overrightarrow{B M}| \\cdot|\\vec{n}|}=\\frac{1}{4}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\n\n\u6545  B M  \u4e0e\u5e73\u9762  V A D  \u6240\u6210\u89d2\u7684\u6b63\u5f26\u503c\u4e3a  \\frac{1}{4}  .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n","protected":false},"excerpt":{"rendered":"<p>\u7a7a\u95f4\u5411\u91cf\u6c42\u957f\u5ea6\u3001\u5939\u89d2<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13],"tags":[],"class_list":["post-5455","post","type-post","status-publish","format-standard","hentry","category-13"],"blocksy_meta":[],"_links":{"self":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts\/5455","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5455"}],"version-history":[{"count":4,"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts\/5455\/revisions"}],"predecessor-version":[{"id":5471,"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts\/5455\/revisions\/5471"}],"wp:attachment":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5455"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5455"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5455"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}