{"id":3500,"date":"2024-11-09T07:20:18","date_gmt":"2024-11-08T23:20:18","guid":{"rendered":"http:\/\/jhwk.online\/?p=3500"},"modified":"2024-11-09T07:20:18","modified_gmt":"2024-11-08T23:20:18","slug":"%e5%b9%b3%e9%9d%a2%e5%90%91%e9%87%8f","status":"publish","type":"post","link":"http:\/\/jhwk.online\/?p=3500","title":{"rendered":"\u5e73\u9762\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-\">\u7528\u57fa\u5e95\u8868\u793a\u5411\u91cf<\/a><\/li><li class=\"\"><a href=\"\/#htoc-111111\">\u5411\u91cf\u5171\u7ebf\u5b9a\u7406<\/a><\/li><li class=\"\"><a href=\"\/#htoc-1111111111111\">\u5e73\u9762\u5411\u91cf\u6570\u91cf\u79ef\u7684\u5e94\u7528<\/a><\/li><\/ul><\/div><\/div>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"htoc-\">\u7528\u57fa\u5e95\u8868\u793a\u5411\u91cf<\/h4>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>1.\u5728  \\triangle A B C  \u4e2d, \u70b9  D, E  \u5206\u522b\u5728\u8fb9  B C, A C  \u4e0a, \u4e14  \\overrightarrow{B D}=2 \\overrightarrow{D C}, \\overrightarrow{C E}=3 \\overrightarrow{E A} , \u82e5  \\overrightarrow{A B}= \\overrightarrow{a}, ~~~~~~~\\\\\\overrightarrow{A C}= \\overrightarrow{b} , \u7528 \\overrightarrow{a},  \\overrightarrow{b} \u8868\u793a  \\overrightarrow{D E} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\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-stackable-columns stk-block-columns stk-block stk-1982b90\" data-block-id=\"1982b90\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-1982b90-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-2e694d7\" data-v=\"4\" data-block-id=\"2e694d7\"><style>@media screen and (min-width:690px){.stk-2e694d7 {flex:var(--stk-flex-grow, 1) 1 66.7\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-frac-1-3-overrightarrow-a-frac-5-12-overrightarrow-b-overrightarrow-d-e-frac-1-3-overrightarrow-b-c-frac-3-4-overrightarrow-c-a-frac-1-3-overrightarrow-a-c-overrightarrow-a-b-frac-3-4-overrightarrow-a-c-frac-1-3-overrightarrow-a-frac-5-12-overrightarrow-b\"><pre>\u7b54\u6848:-\\frac{1}{3} \\overrightarrow{a}-\\frac{5}{12} \\overrightarrow{b} ;\u63d0\u793a:\\overrightarrow{D E} =\\frac{1}{3} \\overrightarrow{B C}+\\frac{3}{4} \\overrightarrow{C A} ~~~~~~~~~\\\\\n=\\frac{1}{3}(\\overrightarrow{A C}-\\overrightarrow{A B})-\\frac{3}{4} \\overrightarrow{A C} \n=-\\frac{1}{3} \\overrightarrow{a}-\\frac{5}{12} \\overrightarrow{b} .~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-e218556\" data-v=\"4\" data-block-id=\"e218556\"><style>@media screen and (min-width:690px){.stk-e218556 {flex:var(--stk-flex-grow, 1) 1 33.3\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/11\/pmxl-01.png\" alt=\"\" class=\"wp-image-3514\" style=\"width:169px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/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.\u5728\u77e9\u5f62  A B C D  \u4e2d,  E  \u4e3a  A B  \u8fb9\u7684\u4e2d\u70b9, \u7ebf\u6bb5  A C  \u548c  D E  \u4ea4\u4e8e\u70b9  F , \u5219  \\overrightarrow{B F}=(\\quad).~~~~~~~~~~~~~~~~~~\\\\\nA.  -\\frac{1}{3} \\overrightarrow{A B}+\\frac{2}{3} \\overrightarrow{A D}\\quad\nB.  \\frac{1}{3} \\overrightarrow{A B}-\\frac{2}{3} \\overrightarrow{A D}\\quad\nC.  \\frac{2}{3} \\overrightarrow{A B}-\\frac{1}{3} \\overrightarrow{A D} \\quad\nD.  -\\frac{2}{3} \\overrightarrow{A B}+\\frac{1}{3} \\overrightarrow{A D} ~~~~~~~~~~~~~~~<\/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-11\"><\/p>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-d9f0659\" data-block-id=\"d9f0659\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-d9f0659-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-f01da68\" data-v=\"4\" data-block-id=\"f01da68\"><style>@media screen and (min-width:690px){.stk-f01da68 {flex:var(--stk-flex-grow, 1) 1 calc(72\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-a-triangle-a-ef-sim-triangle-cdf-rightarrow-af-cf-1-2-overrightarrow-bf-overrightarrow-af-overrightarrow-ab-frac-1-3-overrightarrow-ac-overrightarrow-ab-frac-1-3-overrightarrow-a-b-frac-2-3-overrightarrow-a-d\"><pre>\u7b54\u6848:A;\u63d0\u793a: \\triangle A EF\\sim  \\triangle CDF \\Rightarrow |AF|:|CF|=1:2,~\\\\\n  \\overrightarrow{BF}=  \\overrightarrow{AF}-  \\overrightarrow{AB}=\\frac{1}{3}\\overrightarrow{AC}-  \\overrightarrow{AB}=-\\frac{1}{3} \\overrightarrow{A B}+\\frac{2}{3} \\overrightarrow{A D}.~~~~~~~~~~~~~<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-feaea69\" data-v=\"4\" data-block-id=\"feaea69\"><style>@media screen and (min-width:690px){.stk-feaea69 {flex:var(--stk-flex-grow, 1) 1 calc(28\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/11\/pmxl-02.png\" alt=\"\" class=\"wp-image-3516\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/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>3.\u5982\u56fe, \u7b49\u8170\u68af\u5f62  A B C D  \u4e2d,  A B=B C=C D=3 A D , \u70b9  E  \u4e3a\u7ebf\u6bb5  C D  \u4e0a\u9760\u8fd1  C  \u7684\u4e09\u7b49~~~\\\\\u5206\u70b9, \u70b9  F  \u4e3a\u7ebf\u6bb5  B C  \u7684\u4e2d\u70b9, \u5219  \\overrightarrow{F E}=(\\quad).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA.  -\\frac{11}{18} \\overrightarrow{A B}+\\frac{5}{18} \\overrightarrow{A C} \\quad\nB.  -\\frac{11}{18} \\overrightarrow{A B}+\\frac{11}{9} \\overrightarrow{A C} C.  -\\frac{11}{18} \\overrightarrow{A B}+\\frac{4}{9} \\overrightarrow{A C} \\quad\nD.  -\\frac{1}{2} \\overrightarrow{A B}+\\frac{5}{6} \\overrightarrow{A C} ~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-3e319d3\" data-block-id=\"3e319d3\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-3e319d3-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-e04b96a\" data-v=\"4\" data-block-id=\"e04b96a\"><style>@media screen and (min-width:690px){.stk-e04b96a {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-111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-a-overrightarrow-f-e-overrightarrow-f-c-overrightarrow-c-e-frac-1-2-overrightarrow-b-c-frac-1-3-overrightarrow-c-d-frac-1-2-overrightarrow-a-c-overrightarrow-a-b-frac-1-3-left-overrightarrow-ad-overrightarrow-ac-right-frac-11-18-overrightarrow-a-b-frac-5-18-overrightarrow-a-c\"><pre>\u7b54\u6848: A;\n\u63d0\u793a:\n\n\\overrightarrow{F E}=\\overrightarrow{F C}+\\overrightarrow{C E}=\\frac{1}{2} \\overrightarrow{B C}+\\frac{1}{3} \\overrightarrow{C D}=~~~\\\\\\frac{1}{2}(\\overrightarrow{A C}-\\overrightarrow{A B})+\\frac{1}{3}\\left(\\overrightarrow{AD}-\\overrightarrow{AC}\\right) =-\\frac{11}{18} \\overrightarrow{A B}+\\frac{5}{18} \\overrightarrow{A C}.~\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-4ef48a2\" data-v=\"4\" data-block-id=\"4ef48a2\"><style>@media screen and (min-width:690px){.stk-4ef48a2 {flex:var(--stk-flex-grow, 1) 1 33.3\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/11\/pmxl-1.png\" alt=\"\" class=\"wp-image-3505\" style=\"width:167px;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. \u5728\u5e73\u884c\u56db\u8fb9\u5f62  A B C D  \u4e2d,  A C  \u4e0e  B D  \u4ea4\u4e8e\u70b9  O, F  \u662f\u7ebf\u6bb5  D C  \u4e0a\u7684\u70b9.\u82e5  D C=   3 D F , ~~~~~~~~~\\\\\u8bbe  \\overrightarrow{A C}=\\overrightarrow{a}, \\overrightarrow{B D}=\\overrightarrow{b} , \u5219  \\overrightarrow{A F}=(\\quad) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA.  \\frac{1}{4} \\overrightarrow{a}+\\frac{1}{2} \\overrightarrow{b} \\quad\nB.  \\frac{2}{2} \\overrightarrow{a}+\\frac{1}{3} \\overrightarrow{b} \\quad\nC.  \\frac{1}{2} \\overrightarrow{a}+\\frac{1}{4} \\overrightarrow{b} \\quad\n\nD.  \\frac{1}{3} \\overrightarrow{a}+\\frac{2}{3} \\overrightarrow{b}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-54780a5\" data-block-id=\"54780a5\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-54780a5-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-562ac6a\" data-v=\"4\" data-block-id=\"562ac6a\"><style>@media screen and (min-width:690px){.stk-562ac6a {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-b-overrightarrow-a-overrightarrow-b-overrightarrow-ab-frac-1-2-left-overrightarrow-a-overrightarrow-b-right-overrightarrow-ad-frac-1-2-left-overrightarrow-a-overrightarrow-b-right-overrightarrow-a-f-frac-2-2-overrightarrow-a-frac-1-3-overrightarrow-b\"><pre>\u7b54\u6848: B;\u63d0\u793a:\u5148\u7528\\overrightarrow{a},\\overrightarrow{b}\u8868\u793a\\overrightarrow{AB}=\\frac{1 }{2} \\left ( \\overrightarrow{a}-\\overrightarrow{b} \\right ),~~~~~\\\\\\overrightarrow{AD}=\\frac{1 }{2} \\left ( \\overrightarrow{a}+\\overrightarrow{b} \\right ), \\overrightarrow{A F}=\\frac{2}{2} \\overrightarrow{a}+\\frac{1}{3} \\overrightarrow{b} .~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/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-13d5ffa\" data-v=\"4\" data-block-id=\"13d5ffa\"><style>@media screen and (min-width:690px){.stk-13d5ffa {flex:var(--stk-flex-grow, 1) 1 33.3\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/11\/pmxl-04.png\" alt=\"\" class=\"wp-image-3532\" style=\"width:196px;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>5.\u5728\u68af\u5f62  A B C D  \u4e2d,  A B \/ \/ C D, A B=2 C D, M, N  \u5206\u522b\u4e3a  C D, B C  \u7684\u4e2d\u70b9.\u82e5  \\overrightarrow{A B}=   ~~~~~~~~~~~\\\\\\lambda \\overrightarrow{A M}+\\mu \\overrightarrow{A N} , \u5219  \\lambda+\\mu  \u7b49\u4e8e(\\quad )~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA.  \\frac{1}{5} \\quad\nB.  \\frac{2}{5} \\quad\nC.  \\frac{3}{5} \\quad\nD.  \\frac{4}{5} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/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-11111\"><\/p>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-d299e77\" data-block-id=\"d299e77\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-d299e77-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-1964520\" data-v=\"4\" data-block-id=\"1964520\"><style>@media screen and (min-width:690px){.stk-1964520 {flex:var(--stk-flex-grow, 1) 1 calc(75\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-d-overrightarrow-a-b-overrightarrow-a-d-overrightarrow-a-m-overrightarrow-a-n-overrightarrow-a-b-overrightarrow-a-n-overrightarrow-a-n-overrightarrow-a-b-frac-1-2-overrightarrow-bc-overrightarrow-a-b-frac-1-2-left-overrightarrow-a-b-overrightarrow-a-d-overrightarrow-dc-right-frac-1-2-overrightarrow-a-d-frac-3-4-overrightarrow-a-b-rightarrow-left-begin-matrix-overrightarrow-a-m-overrightarrow-a-d-displaystyle-frac-1-4-overrightarrow-a-b-amp-overrightarrow-a-n-displaystyle-frac-1-2-overrightarrow-a-d-displaystyle-frac-3-4-overrightarrow-a-b-amp-end-matrix-right-overrightarrow-a-b-frac-8-5-overrightarrow-an-frac-4-5-overrightarrow-a-m\"><pre>\u7b54\u6848: D;\n\u63d0\u793a: \u7528\\overrightarrow{A B},\\overrightarrow{A D}\u8868\u793a\u5411\u91cf\\overrightarrow{A M},\\overrightarrow{A N},\u518d\u89e3\u65b9\u7a0b\u7ec4~~~~~\\\\\u5f97\u5230\\overrightarrow{A B}.\u5148\u6c42 \\overrightarrow{A N}\uff0c\\overrightarrow{A N}=~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\\\\\overrightarrow{A B}+\\frac{1}{2}\\overrightarrow{BC}=\\overrightarrow{A B}+\\frac{1}{2}\\left ( -\\overrightarrow{A B}+\\overrightarrow{A D}+\\overrightarrow{DC} \\right ) \n=~~~~~~~~~~~~~~~~~~~~~~~\\\\\\frac{1}{2}\\overrightarrow{A D}+\\frac{3}{4}  \\overrightarrow{A B},\\Rightarrow \n\\left\\{\\begin{matrix}\n \\overrightarrow{A M}=\\overrightarrow{A D}+\\displaystyle \\frac{1}{4} \\overrightarrow{A B} &amp; \\\\\n  \\overrightarrow{A N}=\\displaystyle \\frac{1}{2}\\overrightarrow{A D} +\\displaystyle \\frac{3}{4} \\overrightarrow{A B} &amp;\n\\end{matrix}\\right.,\u89e3\u65b9\u7a0b\u7ec4\u5f97:~~~~~~\\\\\\overrightarrow{A B}= \\frac{8}{5} \\overrightarrow{AN} - \\frac{4}{5} \\overrightarrow{A M} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-a568d3a\" data-v=\"4\" data-block-id=\"a568d3a\"><style>@media screen and (min-width:690px){.stk-a568d3a {flex:var(--stk-flex-grow, 1) 1 calc(25\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/11\/pmxl-4.png\" alt=\"\" class=\"wp-image-3525\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\n<\/details>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"htoc-111111\">\u5411\u91cf\u5171\u7ebf\u5b9a\u7406<\/h4>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>1.\u5df2\u77e5A(x_1,y_1),B(x_2,y_2)\u662f\u5e73\u9762\u4e2d\u7684\u4e24\u4e2a\u4e0d\u540c\u70b9\uff0c\\overrightarrow{AC} =\\lambda\\overrightarrow{CB} ,\u6c42\u70b9C\u7684\u5750\u6807.~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-5cec5e1\" data-block-id=\"5cec5e1\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-5cec5e1-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-476348b\" data-v=\"4\" data-block-id=\"476348b\"><style>@media screen and (min-width:690px){.stk-476348b {flex:var(--stk-flex-grow, 1) 1 calc(72.4\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-left-frac-x-1-lambda-x-2-1-lambda-frac-y-1-lambda-y-2-1-lambda-right-c-overrightarrow-oc-overrightarrow-oc-overrightarrow-oa-overrightarrow-ac-overrightarrow-oa-frac-lambda-1-lambda-overrightarrow-ab-left-frac-x-1-lambda-x-2-1-lambda-frac-y-1-lambda-y-2-1-lambda-right\"><pre>\u7b54\u6848 : \\left ( \\frac{x_1+\\lambda x_2}{1+\\lambda } ,\\frac{y_1+\\lambda y_2}{1+\\lambda } \\right ) ;\u63d0\u793a: C \u70b9\u5750\u6807\u7b49\u4e8e~~~~~~~~~~~~~\\\\\\overrightarrow {OC}\u7684\u5750\u6807,\u5373\\overrightarrow {OC}=\\overrightarrow {OA}+\\overrightarrow {AC}=\\overrightarrow {OA}+\\frac{\\lambda }{1+\\lambda } \\overrightarrow {AB}=~~~~~~~~~~~\\\\\\left ( \\frac{x_1+\\lambda x_2}{1+\\lambda } ,\\frac{y_1+\\lambda y_2}{1+\\lambda } \\right ).~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/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-24aedd2\" data-v=\"4\" data-block-id=\"24aedd2\"><style>@media screen and (min-width:690px){.stk-24aedd2 {flex:var(--stk-flex-grow, 1) 1 calc(27.6\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/11\/pmxl7.png\" alt=\"\" class=\"wp-image-3606\" style=\"width:186px;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.(1)\u8bbe \\boldsymbol{a}  \u4e0e\\boldsymbol{b}  \u662f\u4e24\u4e2a\u4e0d\u5171\u7ebf\u5411\u91cf, \u4e14\u5411\u91cf  \\boldsymbol{a}+\\lambda \\boldsymbol{b}  \u4e0e  -(\\boldsymbol{b}-2\\boldsymbol{a})  \u5171\u7ebf, \u6c42  \\lambda.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n(2)\u8bbe\\boldsymbol{  e_{1} } \u4e0e \\boldsymbol{  e_{2} }  \u662f\u4e24\u4e2a\u4e0d\u5171\u7ebf\u5411\u91cf,  \\overrightarrow{A B}=3 \\boldsymbol{  e_{1} }+2 \\boldsymbol{  e_{2} } , \\overrightarrow{C B}=k \\boldsymbol{  e_{1} }+\\boldsymbol{  e_{2} } , \\overrightarrow{C D}=3 \\boldsymbol{  e_{1} }-2 k \\boldsymbol{  e_{2} }  ,~~~\\\\\u82e5  A, B, D  \u4e09\u70b9\u5171\u7ebf, \u6c42  k  \u7684\u503c.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\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-11111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-1-frac-1-2-boldsymbol-a-lambda-boldsymbol-b-k-2-boldsymbol-a-boldsymbol-b-k-frac-1-2-lambda-frac-1-2-2-frac-9-4-a-b-d-lambda-overrightarrow-a-b-lambda-overrightarrow-b-d-3-e-1-2-e-2-lambda-3-k-e-1-lambda-2-k-1-e-2-left-begin-array-l-3-lambda-3-k-2-lambda-2-k-1-end-array-right-k-frac-9-4\"><pre>\u7b54\u6848: (1)-\\frac{1}{2} ;\u63d0\u793a: \u8bbe  \\boldsymbol{a}+\\lambda \\boldsymbol{b}=k(2 \\boldsymbol{a}   -\\boldsymbol{b}) , \u5f97  k=\\frac{1}{2}, \\lambda=-\\frac{1}{2} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\\\\n(2)-\\frac{9}{4} ;\n\u63d0\u793a:\u7531\u9898\u610f\uff0c  A, B, D \u4e09 \u70b9\u5171\u7ebf\uff0c\u6545\u5fc5\u5b58\u5728\u4e00\u4e2a\u5b9e\u6570  \\lambda  \uff0c\u4f7f\u5f97  \\overrightarrow{A B}=\\lambda \\overrightarrow{B D} .~~~~~\\\\\u6240\u4ee53 e_{1}+2 e_{2}=\\lambda(3-k) e_{1}-\\lambda(2 k+1) e_{2} ,\u6240\u4ee5  \\left\\{\\begin{array}{l}3=\\lambda(3-k), \\\\ 2=-\\lambda(2 k+1)\\end{array}\\right. , \u89e3\u5f97  k=-\\frac{9}{4} .~~~~~~\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>3.\u5982\u56fe\u6240\u793a, \u5728  \\triangle A B C  \u4e2d, \u70b9  O  \u662f  B C  \u7684\u4e2d\u70b9, \u8fc7\u70b9  O  \u7684\u76f4\u7ebf\u5206\u522b\u4ea4  A B, A C  \u6240\u5728\u76f4\u7ebf\u4e8e\u4e0d\\\\\u540c\u7684\u4e24\u70b9  M, N , \u82e5  \\overrightarrow{A B}=m \\overrightarrow{A M}, \\overrightarrow{A C}   =n \\overrightarrow{A N} , \u6c42  m+n  \u7684\u503c.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-784022c\" data-block-id=\"784022c\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-784022c-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-480265c\" data-v=\"4\" data-block-id=\"480265c\"><style>@media screen and (min-width:690px){.stk-480265c {flex:var(--stk-flex-grow, 1) 1 calc(75\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-2-a-o-overrightarrow-a-o-frac-1-2-overrightarrow-a-b-overrightarrow-a-c-frac-m-2-overrightarrow-a-m-frac-n-2-overrightarrow-a-n-m-o-n-frac-m-2-frac-n-2-1-m-n-2\"><pre>\u7b54\u6848: 2;\n\u63d0\u793a:\u8fde\u63a5  A O , \u5219  \\overrightarrow{A O}=\\frac{1}{2}(\\overrightarrow{A B}+\\overrightarrow{A C})=~~~~~~~~~~~~~~~~~~~\\\\\\frac{m}{2} \\overrightarrow{A M}+\\frac{n}{2} \\overrightarrow{A N} , \u56e0\u4e3a  M, O, N  \u70b9\u5171\u7ebf,\u6240\u4ee5  \\frac{m}{2}+\\frac{n}{2}=1 ,~~~~~~~~~~~ \\\\\u6240\u4ee5  m+n=2 .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\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-d508453\" data-v=\"4\" data-block-id=\"d508453\"><style>@media screen and (min-width:690px){.stk-d508453 {flex:var(--stk-flex-grow, 1) 1 calc(25\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/11\/pmxl-8.png\" alt=\"\" class=\"wp-image-3540\" style=\"width:154px;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. \u5df2\u77e5\u70b9M  \u4e3a  \\triangle A B C  \u4e2d  B C  \u8fb9\u4e0a\u7684\u4e2d\u70b9, \u70b9  N  \u6ee1\u8db3  \\overrightarrow{A N}=\\frac{1}{5} \\overrightarrow{A M} , \u8fc7\u70b9N  \u7684\u76f4\u7ebf\u4e0e  A B, ~~~~~\\\\A C  \u5206\u522b\u4ea4\u4e8e  P, Q  \u4e24\u70b9, \u4e14\u8bbe  \\overrightarrow{A P}=x \\overrightarrow{A B}, \\overrightarrow{A Q}=y \\overrightarrow{A C} , \u6c42  \\frac{1}{x}+\\frac{1}{y}  \u7684\u503c.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-e5bddbd\" data-block-id=\"e5bddbd\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-e5bddbd-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-c212200\" data-v=\"4\" data-block-id=\"c212200\"><style>@media screen and (min-width:690px){.stk-c212200 {flex:var(--stk-flex-grow, 1) 1 calc(69.7\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-10-overrightarrow-a-n-frac-1-5-overrightarrow-a-m-frac-1-10-overrightarrow-a-b-overrightarrow-a-c-frac-1-10-left-frac-1-x-overrightarrow-a-p-frac-1-y-overrightarrow-a-q-right-because-p-n-q-therefore-frac-1-10-x-frac-1-10-y-1-frac-1-x-frac-1-y-10\"><pre>\u7b54\u6848:10;\u89e3\u6790:\n \\overrightarrow{A N}=\\frac{1}{5} \\overrightarrow{A M}=\\frac{1}{10}(\\overrightarrow{A B}+\\overrightarrow{A C})=~~~~~~~~~~~\\\\\\frac{1}{10}\\left(\\frac{1}{x} \\overrightarrow{A P}+\\frac{1}{y} \\overrightarrow{A Q}\\right),\\because P, N, Q \u4e09\u70b9\u5171\u7ebf\uff0c~~~~~~~~~~~~~~~~~~~~~\\\\  \\therefore \\frac{1}{10 x}+\\frac{1}{10 y}=1 , \u5373  \\frac{1}{x}+\\frac{1}{y}=10 .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/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-5419b46\" data-v=\"4\" data-block-id=\"5419b46\"><style>@media screen and (min-width:690px){.stk-5419b46 {flex:var(--stk-flex-grow, 1) 1 calc(30.299999999999997\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/11\/pmxl-9.png\" alt=\"\" class=\"wp-image-3543\" style=\"width:179px;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>5.\u5982\u56fe,  \\triangle A B C  \u4e2d, \u70b9  M  \u662f  B C  \u7684\u4e2d\u70b9, \u70b9  N  \u6ee1\u8db3  \\overrightarrow{A N}=\\frac{2}{3} \\overrightarrow{A B}, A M  \u4e0e  C N \u4ea4\u4e8e\u70b9  D, \\overrightarrow{A D}=\\\\\\lambda \\overrightarrow{A M} , \u5219  \\lambda  \u7b49\u4e8e (\\quad )~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA.  \\frac{2}{3} \\quad\nB.  \\frac{3}{4} \\quad\nC.  \\frac{4}{5} \\quad\nD.  \\frac{5}{6} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-8055990\" data-block-id=\"8055990\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-8055990-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-5fe234b\" data-v=\"4\" data-block-id=\"5fe234b\"><style>@media screen and (min-width:690px){.stk-5fe234b {flex:var(--stk-flex-grow, 1) 1 calc(75\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-c-overrightarrow-a-d-lambda-overrightarrow-a-m-frac-lambda-2-overrightarrow-a-b-frac-lambda-2-overrightarrow-a-c-frac-3-lambda-4-overrightarrow-a-n-frac-lambda-2-overrightarrow-a-c-c-d-n-frac-3-lambda-4-frac-lambda-2-1-lambda-frac-4-5\"><pre>\u7b54\u6848: C;\n\u63d0\u793a:\\overrightarrow{A D}=\\lambda \\overrightarrow{A M}=\\frac{\\lambda}{2} \\overrightarrow{A B}+\\frac{\\lambda}{2} \\overrightarrow{A C}=~~~~~~~~~~~~~~~~~~~~~~\\\\\\frac{3 \\lambda}{4} \\overrightarrow{A N}+\\frac{\\lambda}{2} \\overrightarrow{A C} ,\n\n\u56e0\u4e3a\u70b9  C, D, N  \u5171\u7ebf, \u5219\u6709  \\frac{3 \\lambda}{4}+\\frac{\\lambda}{2}=1 , ~~~~~~~~~~~\\\\\u89e3\u5f97  \\lambda=\\frac{4}{5} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/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-bbbb199\" data-v=\"4\" data-block-id=\"bbbb199\"><style>@media screen and (min-width:690px){.stk-bbbb199 {flex:var(--stk-flex-grow, 1) 1 calc(25\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/11\/pmxl-05.png\" alt=\"\" class=\"wp-image-3556\" style=\"width:154px;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>6.\u5728  \\triangle A B C  \u4e2d, \u70b9  P  \u662f  A B  \u4e0a\u4e00\u70b9, \u4e14  \\overrightarrow{C P}=\\frac{2}{3} \\overrightarrow{C A}+\\frac{1}{3} \\overrightarrow{C B}, Q  \u662f  B C  \u7684\u4e2d\u70b9,  A Q \u4e0e  C P  \u7684~~\\\\\u4ea4\u70b9\u4e3a  M , \u53c8  \\overrightarrow{C M}=t \\overrightarrow{C P} , \u6c42  t  \u7684\u503c.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-c0ce831\" data-block-id=\"c0ce831\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-c0ce831-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-1ecb44e\" data-v=\"4\" data-block-id=\"1ecb44e\"><style>@media screen and (min-width:690px){.stk-1ecb44e {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-111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-frac-3-4-a-m-q-rightarrow-overrightarrow-c-m-t-overrightarrow-c-p-t-left-frac-2-3-overrightarrow-c-a-frac-1-3-overrightarrow-c-b-right-frac-2-t-3-overrightarrow-c-a-frac-2-t-3-overrightarrow-c-q-t-frac-3-4\"><pre>\u7b54\u6848  :\\frac{3}{4}; \u63d0\u793a: A, M, Q  \u4e09\u70b9\u5171\u7ebf,\\Rightarrow\n \\overrightarrow{C M}=t \\overrightarrow{C P}=~~~~~~~~~\\\\t\\left(\\frac{2}{3} \\overrightarrow{C A}+\\frac{1}{3} \\overrightarrow{C B}\\right)=\\frac{2 t}{3} \\overrightarrow{C A}+\\frac{2 t}{3} \\overrightarrow{C Q} , \u5f97  t=\\frac{3}{4} . ~~~~~~~~~~~~~~~~~~~<\/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-0d5ab48\" data-v=\"4\" data-block-id=\"0d5ab48\"><style>@media screen and (min-width:690px){.stk-0d5ab48 {flex:var(--stk-flex-grow, 1) 1 calc(29.5\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/11\/pmxl-10.png\" alt=\"\" class=\"wp-image-3546\" style=\"width:176px;height:auto\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"htoc-1111111111111\">\u5e73\u9762\u5411\u91cf\u6570\u91cf\u79ef\u7684\u5e94\u7528<\/h4>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\"><pre>1.\u5df2\u77e5\u5411\u91cf  \\boldsymbol{a}, \\boldsymbol{b}  \u6ee1\u8db3  |\\boldsymbol{a}|=1,|\\boldsymbol{b}|=2, \\boldsymbol{a}-\\boldsymbol{b}=(\\sqrt{3}, \\sqrt{2}) , \u5219  |2 \\boldsymbol{a}-\\boldsymbol{b}|  \u7b49\u4e8e  (\\quad) ~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA.  2 \\sqrt{2} \\quad\nB.  \\sqrt{17} \\quad\nC.  \\sqrt{15} \\quad\nD.  2 \\sqrt{5} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/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-11111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-a-boldsymbol-a-cdot-boldsymbol-b-0-2-boldsymbol-a-boldsymbol-b-2-8\"><pre>\u7b54\u6848:A;\u63d0\u793a: \\boldsymbol{a} \\cdot \\boldsymbol{b} =0\uff0c(2 \\boldsymbol{a}-\\boldsymbol{b})^2=8.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/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.\u5df2\u77e5\u975e\u96f6\u5411\u91cf  \\boldsymbol{a}, \\boldsymbol{b}  \u6ee1\u8db3  |\\boldsymbol{a}|=2|\\boldsymbol{b}| , \u4e14  (\\boldsymbol{a}-\\boldsymbol{b}) \\perp \\boldsymbol{b} , \u5219  \\boldsymbol{a}  \u4e0e  \\boldsymbol{b}  \u7684\u5939\u89d2\u4e3a( \\quad)~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA.  \\frac{\\pi}{6} \\quad\nB.  \\frac{\\pi}{3} \\quad\nC.  \\frac{2 \\pi}{3} \\quad\nD.  \\frac{5 \\pi}{6} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/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-111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-b-boldsymbol-a-boldsymbol-b-cdot-boldsymbol-b-0-therefore-boldsymbol-a-cdot-boldsymbol-b-boldsymbol-b-2-because-boldsymbol-a-2-boldsymbol-b-therefore-cos-langle-boldsymbol-a-boldsymbol-b-rangle-frac-boldsymbol-a-cdot-boldsymbol-b-boldsymbol-a-cdot-boldsymbol-b-frac-boldsymbol-b-2-2-boldsymbol-b-2-frac-1-2-because-0-leqslant-langle-boldsymbol-a-boldsymbol-b-rangle-leqslant-pi-therefore-boldsymbol-a-boldsymbol-b-frac-pi-3\"><pre>\u7b54\u6848:B;\u63d0\u793a:  (\\boldsymbol{a}-\\boldsymbol{b}) \\cdot \\boldsymbol{b}=0, \\therefore \\boldsymbol{a} \\cdot \\boldsymbol{b}=\\boldsymbol{b}^{2} . \\because|\\boldsymbol{a}|=2|\\boldsymbol{b}|, \\therefore \\cos \\langle\\boldsymbol{a}, \\boldsymbol{b}\\rangle=\\frac{\\boldsymbol{a} \\cdot \\boldsymbol{b}}{|\\boldsymbol{a}| \\cdot|\\boldsymbol{b}|}   =~~~~~~~\\\\\\frac{\\boldsymbol{b}^{2}}{2 \\boldsymbol{b}^{2}}=\\frac{1}{2} \\because 0 \\leqslant\\langle \\boldsymbol{a}, \\boldsymbol{b}\\rangle \\leqslant \\pi, \\therefore \\boldsymbol{a}  \u4e0e  \\boldsymbol{b}  \u7684\u5939\u89d2\u4e3a  \\frac{\\pi}{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>3.\u5df2\u77e5\u5411\u91cf  \\boldsymbol{a}, \\boldsymbol{b}  \u6ee1\u8db3  |\\boldsymbol{a}|=5,|\\boldsymbol{b}|=6, \\boldsymbol{a} \\cdot \\boldsymbol{b}=-6 , \u5219  \\cos \\langle\\boldsymbol{a}, \\boldsymbol{a}+\\boldsymbol{b}\\rangle=(\\quad)~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\\\\nA.  -\\frac{31}{35} \\quad\nB.  -\\frac{19}{35} \\quad\nC.  \\frac{17}{35} \\quad\nD.  \\frac{19}{35} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/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-1111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-d-boldsymbol-a-boldsymbol-b-2-boldsymbol-a-boldsymbol-b-2-49-therefore-boldsymbol-a-boldsymbol-b-7-therefore-cos-langle-boldsymbol-a-boldsymbol-a-boldsymbol-b-rangle-frac-19-35\"><pre>\u7b54\u6848:D;\u63d0\u793a:  |\\boldsymbol{a}+\\boldsymbol{b}|^{2}=(\\boldsymbol{a}+\\boldsymbol{b})^{2}=49, \\therefore|\\boldsymbol{a}+\\boldsymbol{b}|=7, \\therefore \\cos \\langle\\boldsymbol{a}, \\boldsymbol{a}+\\boldsymbol{b}\\rangle=\\frac{19}{35} .~~~~~~~~~~~~~~\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>4.\u5982\u56fe, \u5728\u8fb9\u957f\u4e3a 2 \u7684\u7b49\u8fb9  \\triangle A B C  \u4e2d, \u70b9  E  \u4e3a\u4e2d\u7ebf  B D  \u7684\u4e09\u7b49\u5206\u70b9(\u9760\u8fd1\u70b9  B  ),\u70b9  F  \u4e3a  B C  ~~~~\\\\\u7684\u4e2d\u70b9, \u5219  \\overrightarrow{F E} \\cdot \\overrightarrow{F C}=(\\quad) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\nA.  -\\frac{\\sqrt{3}}{4} \\quad\nB.  -\\frac{1}{2} \\quad\nC.  \\frac{3}{4} \\quad\nD.  \\frac{1}{2} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-068414b\" data-block-id=\"068414b\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-068414b-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-083d691\" data-v=\"4\" data-block-id=\"083d691\"><style>@media screen and (min-width:690px){.stk-083d691 {flex:var(--stk-flex-grow, 1) 1 calc(71.6\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-b-f-overrightarrow-f-e-left-frac-1-2-frac-sqrt-3-6-right-overrightarrow-f-c-left-frac-1-2-frac-sqrt-3-2-right-therefore-overrightarrow-f-e-cdot-overrightarrow-f-c-frac-1-2-overrightarrow-bc-overrightarrow-m-overrightarrow-ba-overrightarrow-n-overrightarrow-f-e-cdot-overrightarrow-f-c-frac-1-2-vec-m-cdot-left-frac-1-6-vec-n-frac-1-3-vec-m-right-frac-1-2\"><pre>\u7b54\u6848:B;\u63d0\u793a:\n\u6cd5\u4e00\u3001 \u4ee5F\u4e3a\u539f\u70b9\u5efa\u7acb\u5750\u6807\u7cfb\uff0c~~~~~~~~~~~~~~~~~\\\\  \\overrightarrow{F E}=\\left(\\frac{1}{2}, \\frac{\\sqrt{3}}{6}\\right), \\overrightarrow{F C}=\\left(-\\frac{1}{2},-\\frac{\\sqrt{3}}{2}\\right) ,\n \\therefore \\overrightarrow{F E} \\cdot \\overrightarrow{F C}=-\\frac{1}{2} .\\\\\n\u6cd5\u4e8c \u3001\u8f6c\u5316\u4e3a\u57fa\u5e95\u7684\u8fd0\u7b97\uff0c\u8bbe\\overrightarrow{BC} =\\overrightarrow{m}, \\overrightarrow{BA} =\\overrightarrow{n}, ~~~~~~~~~~~~~~\\\\\\overrightarrow{F E} \\cdot \\overrightarrow{F C}=\\frac{1}{2} \\vec{m} \\cdot\\left(\\frac{1}{6} \\vec{n}-\\frac{1}{3} \\vec{m}\\right)=-\\frac{1}{2} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\\\n\u2605\u2605\u6570\u91cf\u79ef\u7684\u8fd0\u7b97\uff0c\u901a\u5e38\u8f6c\u5316\u4e3a\u57fa\u5e95\u8fd0\u7b97\u6216\u5750\u6807\u8fd0\u7b97.~~~~~~<\/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-43ec0d1\" data-v=\"4\" data-block-id=\"43ec0d1\"><style>@media screen and (min-width:690px){.stk-43ec0d1 {flex:var(--stk-flex-grow, 1) 1 calc(28.4\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/11\/pmxl10.png\" alt=\"\" class=\"wp-image-3617\" style=\"width:160px;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>5. \u5df2\u77e5\u5411\u91cf  \\boldsymbol{a}, \\boldsymbol{b}, \\boldsymbol{c}  \u6ee1\u8db3  |\\boldsymbol{a}|=|\\boldsymbol{b}|=1,|\\boldsymbol{c}|=\\sqrt{2} , \u4e14  \\boldsymbol{a}+\\boldsymbol{b}+\\boldsymbol{c}=\\mathbf{0} , \u6c42  \\cos \\langle\\boldsymbol{a}-\\boldsymbol{c}, \\boldsymbol{b}   -\\boldsymbol{c}\\rangle.~~~~~~<\/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-111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-bfb41a2\" data-block-id=\"bfb41a2\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-bfb41a2-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-2f76d2c\" data-v=\"4\" data-block-id=\"2f76d2c\"><style>@media screen and (min-width:690px){.stk-2f76d2c {flex:var(--stk-flex-grow, 1) 1 calc(75\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-frac-4-5-boldsymbol-c-boldsymbol-a-boldsymbol-b-boldsymbol-c-2-boldsymbol-a-2-boldsymbol-b-2-2-boldsymbol-a-cdot-boldsymbol-b-boldsymbol-a-cdot-boldsymbol-b-0-therefore-boldsymbol-a-boldsymbol-c-2-1-boldsymbol-b-boldsymbol-c-1-2-cos-langle-boldsymbol-a-boldsymbol-c-boldsymbol-b-boldsymbol-c-rangle-frac-4-5\"><pre>\u7b54\u6848: \\frac{4}{5} \n\n;\u63d0\u793a: -\\boldsymbol{c}=\\boldsymbol{a}+\\boldsymbol{b},  \\boldsymbol{c}^{2}=\\boldsymbol{a}^{2}+\\boldsymbol{b}^{2}+2 \\boldsymbol{a} \\cdot \\boldsymbol{b} , \u89e3\u5f97~~~~~~~~~~\\\\  \\boldsymbol{a} \\cdot \\boldsymbol{b}=0 . \u5efa\u7cfb\u5982\u56fe,  \\therefore \\boldsymbol{a}-\\boldsymbol{c}=(2,1), \\boldsymbol{b}-\\boldsymbol{c}=(1,2) ,\u5219 ~~~~~~~~~~~~ \\\\\\cos \\langle\\boldsymbol{a}-\\boldsymbol{c}, \\boldsymbol{b}-\\boldsymbol{c}\\rangle=\\frac{4}{5} .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-a7f8978\" data-v=\"4\" data-block-id=\"a7f8978\"><style>@media screen and (min-width:690px){.stk-a7f8978 {flex:var(--stk-flex-grow, 1) 1 calc(25\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/11\/pmxl12.png\" alt=\"\" class=\"wp-image-3625\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/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. \u5728\u5e73\u9762\u76f4\u89d2\u5750\u6807\u7cfb\u4e2d,  O  \u4e3a\u539f\u70b9,  A(-1,0), B(0, \\sqrt{3}), C(3,0) , \u52a8\u70b9  D  \u6ee1\u8db3  |\\overrightarrow{C D}|=1 , \u6c42 ~\\\\ |\\overrightarrow{O A}+\\overrightarrow{O B}+\\overrightarrow{O D}|  \u7684\u6700\u5927\u503c\u662f.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\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-111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-295bca8\" data-block-id=\"295bca8\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-295bca8-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-81a3fe5\" data-v=\"4\" data-block-id=\"81a3fe5\"><style>@media screen and (min-width:690px){.stk-81a3fe5 {flex:var(--stk-flex-grow, 1) 1 calc(77.9\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-1-sqrt-7-d-x-y-overrightarrow-c-d-1-x-3-2-y-2-1-overrightarrow-o-a-overrightarrow-o-b-overrightarrow-o-d-x-1-y-sqrt-3-overrightarrow-o-a-overrightarrow-o-b-overrightarrow-o-d-sqrt-x-1-2-y-sqrt-3-2-1-sqrt-3-x-3-2-y-2-1-e-c-1-sqrt-7-1\"><pre>\u7b54\u6848 : 1+\\sqrt{7}; \n\u63d0\u793a: \u8bbe  D(x, y) ,   |\\overrightarrow{C D}|=1 \\Rightarrow  (x-3)^{2}+y^{2}=1 ,\n\\\\\u5219  \\overrightarrow{O A}+\\overrightarrow{O B}+\\overrightarrow{O D}=(x-1, y+\\sqrt{3}) ,\n\u6545  |\\overrightarrow{O A}+\\overrightarrow{O B}+\\overrightarrow{O D}|=~~\\\\\\sqrt{(x-1)^{2}+(y+\\sqrt{3})^{2}} , \u51e0\u4f55\u610f\u4e49\u4e3a\u70b9  (1,-\\sqrt{3}) \u5230\u5706 ~~~~~~~~~~~~~~~~~~~~ \\\\(x-3)^{2}+y^{2}=1  \u7684\u8ddd\u79bb, \u6700\u5927\u503c\u4e3a  |E C|+1=\\sqrt{7}+1 .~~~~~~~~~~~~~~~~<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-eab1d82\" data-v=\"4\" data-block-id=\"eab1d82\"><style>@media screen and (min-width:690px){.stk-eab1d82 {flex:var(--stk-flex-grow, 1) 1 calc(22.1\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/11\/pmxl11.png\" alt=\"\" class=\"wp-image-3628\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/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>7.\u5df2\u77e5  \\boldsymbol{a}, \\boldsymbol{b}  \u662f\u5355\u4f4d\u5411\u91cf,  \\boldsymbol{a} \\cdot \\boldsymbol{b}=0 . \u82e5\u5411\u91cf  \\boldsymbol{c}  \u6ee1\u8db3  |\\boldsymbol{c}-\\boldsymbol{a}-\\boldsymbol{b}|=1 , \u6c42  |\\boldsymbol{c}|  \u7684\u6700\u5927\u503c\u662f.~~~~~~~~~~~~~~<\/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-1111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-fe92baa\" data-block-id=\"fe92baa\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-fe92baa-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-3f6f61b\" data-v=\"4\" data-block-id=\"3f6f61b\"><style>@media screen and (min-width:690px){.stk-3f6f61b {flex:var(--stk-flex-grow, 1) 1 calc(75\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-sqrt-2-1-boldsymbol-a-cdot-boldsymbol-b-0-boldsymbol-a-perp-boldsymbol-b-overrightarrow-o-a-boldsymbol-a-1-0-overrightarrow-o-b-boldsymbol-b-0-1-boldsymbol-c-overrightarrow-o-c-x-y-boldsymbol-c-boldsymbol-a-boldsymbol-b-1-x-1-2-y-1-2-1-c-1-1-1-c-text-max-sqrt-2-11\"><pre>\u7b54\u6848: \\sqrt{2}+1 ;\n\u63d0\u793a: \u7531  \\boldsymbol{a} \\cdot \\boldsymbol{b}=0 , \u5f97  \\boldsymbol{a} \\perp \\boldsymbol{b} ,\u5efa\u7acb\u5750\u6807\u7cfb, ~~~~~~~~~~\\\\\u5219  \\overrightarrow{O A}=\\boldsymbol{a}=(1,0), \\overrightarrow{O B}=\\boldsymbol{b}=(0,1) .\n\n\u8bbe  \\boldsymbol{c}=\\overrightarrow{O C}=(x, y) , ~~~~~~~~\\\\\u7531  |\\boldsymbol{c}-\\boldsymbol{a}-\\boldsymbol{b}|=1 ,\n\u5f97  (x-1)^{2}+(y-1)^{2}=1 ,\n\u6240\u4ee5\u70b9  C  \u5728\u4ee5~~~~ \\\\ (1,1)  \u4e3a\u5706\u5fc3, 1 \u4e3a\u534a\u5f84\u7684\u5706\u4e0a.\n\u6240\u4ee5  |c|_{\\text {max }}=\\sqrt{2}+1 .~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-d1cae9e\" data-v=\"4\" data-block-id=\"d1cae9e\"><style>@media screen and (min-width:690px){.stk-d1cae9e {flex:var(--stk-flex-grow, 1) 1 calc(25\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/11\/pmxl13.png\" alt=\"\" class=\"wp-image-3634\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/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>8.\u5982\u56fe, \u5728  \\triangle A B C  \u4e2d,  \\cos \\angle B A C=\\frac{1}{4} , \u70b9,  D  \u5728\u7ebf\u6bb5  B C  \u4e0a, \u4e14  B D=3 D C, A D=   \\frac{\\sqrt{15}}{2} , ~~~~\\\\\u6c42  \\triangle A B C  \u7684\u9762\u79ef\u7684\u6700\u5927\u503c.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-1fb8cd4\" data-block-id=\"1fb8cd4\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-1fb8cd4-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-acf9d68\" data-v=\"4\" data-block-id=\"acf9d68\"><style>@media screen and (min-width:690px){.stk-acf9d68 {flex:var(--stk-flex-grow, 1) 1 calc(75\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-11111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-sqrt-15-overrightarrow-a-d-frac-1-4-overrightarrow-a-b-frac-3-4-overrightarrow-a-c-quad-rightarrow-overrightarrow-a-d-left-frac-1-4-overrightarrow-a-b-frac-3-4-overrightarrow-a-c-right-2-rightarrow-frac-15-4-frac-1-16-c-2-frac-9-16-b-2-frac-3-32-b-c-geqslant-frac-15-32-b-c-c-3-b-b-c-leqslant-8-sin-angle-b-a-c-frac-sqrt-15-4-s-triangle-a-b-c-frac-1-2-b-c-sin-angle-b-a-c-leqslant-sqrt-15\"><pre>\u7b54\u6848 : \\sqrt{15} \u63d0\u793a:\n\\overrightarrow{A D}=\\frac{1}{4} \\overrightarrow{A B}+\\frac{3}{4} \\overrightarrow{A C},   \\Rightarrow~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\\\\\overrightarrow{A D}=\\left(\\frac{1}{4} \\overrightarrow{A B}+\\frac{3}{4} \\overrightarrow{A C}\\right)^{2} \\Rightarrow \\frac{15}{4}=\\frac{1}{16} c^{2}+\\frac{9}{16} b^{2}+\\frac{3}{32} b c \\geqslant \\frac{15}{32} b c,\\\\\n\n\n\u5f53\u4e14\u4ec5\u5f53  c=3 b  \u65f6\uff0c\u7b49\u53f7\u6210\u7acb\u3002\n\u6240\u4ee5  b c \\leqslant 8 , \u53c8~~~~~~~~~~~~~~~~~~~~~~~~~\\\\  \\sin \\angle B A C=\\frac{\\sqrt{15}}{4} , \u6240\u4ee5  S_{\\triangle A B C}=\\frac{1}{2} b c \\sin \\angle B A C \\leqslant \\sqrt{15} .~~~~~~~~\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-b77c5ea\" data-v=\"4\" data-block-id=\"b77c5ea\"><style>@media screen and (min-width:690px){.stk-b77c5ea {flex:var(--stk-flex-grow, 1) 1 calc(25\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/11\/pmxl14.png\" alt=\"\" class=\"wp-image-3643\"\/><\/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>9.\u5728  \\triangle A B C  \u4e2d,  A C=9, \\angle A=60^{\\circ}, D  \u70b9\u6ee1\u8db3  \\overrightarrow{C D}=2 \\overrightarrow{D B}, A D=\\sqrt{37} , \u5219  B C  \u7684\u957f\u4e3a (\\quad )\\\\\nA.  3 \\sqrt{7} \\quad \nB.  3 \\sqrt{6} \\quad \nC.  3 \\sqrt{3} \\quad \nD. 6~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-6af7560\" data-block-id=\"6af7560\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-6af7560-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-17b873d\" data-v=\"4\" data-block-id=\"17b873d\"><style>@media screen and (min-width:690px){.stk-17b873d {flex:var(--stk-flex-grow, 1) 1 calc(75\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-1111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-a-overrightarrow-c-d-2-overrightarrow-d-b-overrightarrow-a-d-overrightarrow-a-b-frac-1-3-overrightarrow-b-c-overrightarrow-a-b-frac-1-3-overrightarrow-a-c-overrightarrow-a-b-frac-2-3-overrightarrow-a-b-frac-1-3-overrightarrow-a-c-a-b-x-overrightarrow-a-d-2-left-frac-2-3-overrightarrow-a-b-frac-1-3-overrightarrow-a-c-right-2-2-x-2-9-x-126-0-x-6-a-b-6-overrightarrow-b-c-3-sqrt-7\"><pre>\u7b54\u6848 :A;\u63d0\u793a:   \\overrightarrow{A D}=\\overrightarrow{A B}+\\frac{1}{3} \\overrightarrow{B C}=\\overrightarrow{A B}+\\frac{1}{3}(\\overrightarrow{A C}-\\overrightarrow{A B})=~\\\\\\frac{2}{3} \\overrightarrow{A B}+\\frac{1}{3} \\overrightarrow{A C} ,\n\u8bbe  A B=x , \u5219  \\overrightarrow{A D}{ }^{2}=\\left(\\frac{2}{3} \\overrightarrow{A B}+\\frac{1}{3} \\overrightarrow{A C}\\right)^{2} , ~~~~~~~~~~\\\\\u5373  2 x^{2}+9 x-126=0 ,\n x=6 , \u5373  A B=6 , \u6240\u4ee5  |\\overrightarrow{B C}|==3 \\sqrt{7} .~~<\/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-e1b3e88\" data-v=\"4\" data-block-id=\"e1b3e88\"><style>@media screen and (min-width:690px){.stk-e1b3e88 {flex:var(--stk-flex-grow, 1) 1 calc(25\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/11\/pmxl15.png\" alt=\"\" class=\"wp-image-3644\"\/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity is-style-wide\"\/>\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>10.\u5df2\u77e5\u70b9  A, B, C  \u5747\u5728\u534a\u5f84\u4e3a  \\sqrt{2}  \u7684\u5706\u4e0a, \u82e5  |A B|=2 , \u6c42  \\overrightarrow{A C} \\cdot \\overrightarrow{B C}  \u7684\u53d6\u503c\u8303\u56f4.~~~~~~~~~~~~~~~~~<\/pre><\/div>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-257d52d\" data-block-id=\"257d52d\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-257d52d-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-b695d02\" data-v=\"4\" data-block-id=\"b695d02\"><style>@media screen and (min-width:690px){.stk-b695d02 {flex:var(--stk-flex-grow, 1) 1 calc(75\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848<\/summary>\n<p id=\"htoc-111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-d1\"><pre>\u7b54\u6848:[-1,3];\u63d0\u793a:A(\\sqrt{2},0 ),B(0,\\sqrt{2} ),\u8bbeC(\\cos \\theta ,\\sin \\theta ),~~~~~~\\\\\\overrightarrow{AC}\\cdot \\overrightarrow{BC}\n=(\\cos\\theta -\\sqrt{2},\\sin\\theta  ) (\\cos\\theta,\\sin\\theta-\\sqrt{2} )=~~~~~~~~~~~~~~~~~~\\\\1-2\\sin(\\theta+\\frac{\\pi }{4} )\n\\in [-1,3].~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/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-39aead7\" data-v=\"4\" data-block-id=\"39aead7\"><style>@media screen and (min-width:690px){.stk-39aead7 {flex:var(--stk-flex-grow, 1) 1 calc(25\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/11\/%E5%B1%8F%E5%B9%95%E6%88%AA%E5%9B%BE-2024-11-14-163412.png\" alt=\"\" class=\"wp-image-3655\" style=\"width:164px;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>11.\u5728  \\triangle A B C  \u4e2d,  A C=3, B C=4, \\angle C=90^{\\circ} ,P  \u4e3a  \\triangle A B C  \u6240\u5728\u5e73\u9762\u5185\u7684\u52a8\u70b9\uff0c\u4e14  PC=1  \uff0c\\\\\n\u6c42  \\overrightarrow{PA} \\cdot \\overrightarrow{PB}  \u7684\u53d6\u503c\u8303\u56f4.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/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-1111111111111111111111111\"><\/p>\n\n\n\n<div class=\"wp-block-stackable-columns stk-block-columns stk-block stk-f62eac9\" data-block-id=\"f62eac9\"><div class=\"stk-row stk-inner-blocks stk-block-content stk-content-align stk-f62eac9-column\">\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-381e73b\" data-v=\"4\" data-block-id=\"381e73b\"><style>@media screen and (min-width:690px){.stk-381e73b {flex:var(--stk-flex-grow, 1) 1 calc(71.8\n<div class=\"wp-block-katex-display-block katex-eq\" data-katex-display=\"true\" id=\"htoc-4-6-a-0-3-b-4-0-p-cos-theta-sin-theta-overrightarrow-pa-cdot-overrightarrow-pb-cos-theta-3-sin-theta-4-cos-theta-sin-theta-1-5-sin-theta-varphi-in-4-6\"><pre>\u7b54\u6848:[-4,6];\u63d0\u793a:\u8bbeA(0,3 ),B(4,0 ),P(\\cos \\theta ,\\sin \\theta ),~~~~~~~~\\\\\\overrightarrow{PA}\\cdot \\overrightarrow{PB}\n=(-\\cos\\theta ,3-\\sin\\theta  ) (4-\\cos\\theta,-\\sin\\theta)=~~~~~~~~~~~~~\\\\1-5\\sin(\\theta+\\varphi   )\n\\in [-4,6].~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/pre><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-stackable-column stk-block-column stk-column stk-block stk-6ed3638\" data-v=\"4\" data-block-id=\"6ed3638\"><style>@media screen and (min-width:690px){.stk-6ed3638 {flex:var(--stk-flex-grow, 1) 1 calc(28.2\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"http:\/\/47.95.222.19\/wp-content\/uploads\/2024\/11\/%E5%B1%8F%E5%B9%95%E6%88%AA%E5%9B%BE-2024-11-14-173705.png\" alt=\"\" class=\"wp-image-3656\"\/><\/figure>\n<\/div><\/div><\/div>\n<\/div><\/div>\n<\/details>\n\n\n\n \n","protected":false},"excerpt":{"rendered":"<p>\u7528\u57fa\u5e95\u8868\u793a\u5411\u91cf \u5411\u91cf\u5171\u7ebf\u5b9a\u7406 \u5e73\u9762\u5411\u91cf\u6570\u91cf\u79ef\u7684\u5e94\u7528<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-3500","post","type-post","status-publish","format-standard","hentry","category-2"],"blocksy_meta":[],"_links":{"self":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts\/3500","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=3500"}],"version-history":[{"count":0,"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts\/3500\/revisions"}],"wp:attachment":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3500"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3500"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3500"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}