{"id":211,"date":"2023-07-11T19:16:54","date_gmt":"2023-07-11T19:16:54","guid":{"rendered":"https:\/\/mathority.org\/cn\/%e8%ae%a1%e7%ae%97%e4%b8%a4%e4%b8%aa%e5%90%91%e9%87%8f%e4%b9%8b%e9%97%b4%e7%9a%84%e6%a0%87%e9%87%8f%e7%a7%af-%e5%b7%b2%e8%a7%a3%e5%86%b3%e7%9a%84%e7%a4%ba%e4%be%8b%e7%bb%83%e4%b9%a0\/"},"modified":"2023-07-11T19:16:54","modified_gmt":"2023-07-11T19:16:54","slug":"%e8%ae%a1%e7%ae%97%e4%b8%a4%e4%b8%aa%e5%90%91%e9%87%8f%e4%b9%8b%e9%97%b4%e7%9a%84%e6%a0%87%e9%87%8f%e7%a7%af-%e5%b7%b2%e8%a7%a3%e5%86%b3%e7%9a%84%e7%a4%ba%e4%be%8b%e7%bb%83%e4%b9%a0","status":"publish","type":"post","link":"https:\/\/mathority.org\/cn\/%e8%ae%a1%e7%ae%97%e4%b8%a4%e4%b8%aa%e5%90%91%e9%87%8f%e4%b9%8b%e9%97%b4%e7%9a%84%e6%a0%87%e9%87%8f%e7%a7%af-%e5%b7%b2%e8%a7%a3%e5%86%b3%e7%9a%84%e7%a4%ba%e4%be%8b%e7%bb%83%e4%b9%a0\/","title":{"rendered":"\u8ba1\u7b97\u4e24\u4e2a\u5411\u91cf\u7684\u70b9\u79ef"},"content":{"rendered":"<p>\u5728\u6b64\u9875\u9762\u4e0a\uff0c\u60a8\u5c06\u4e86\u89e3\u5b83\u662f\u4ec0\u4e48\u4ee5\u53ca\u5982\u4f55\u8ba1\u7b97\u4e24\u4e2a\u5411\u91cf\u7684\u70b9\u79ef\u3002\u60a8\u8fd8\u5c06\u5b66\u4e60\u5982\u4f55\u4f7f\u7528\u70b9\u79ef\u6c42\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u89d2\u5ea6\uff0c\u6b64\u5916\u8fd8\u5b66\u4e60\u70b9\u79ef\u7684\u6240\u6709\u5c5e\u6027\u3002\u6700\u540e\uff0c\u60a8\u5c06\u80fd\u591f\u901a\u8fc7\u9010\u6b65\u89e3\u51b3\u7684\u793a\u4f8b\u548c\u7ec3\u4e60\u8fdb\u884c\u7ec3\u4e60\u3002 <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-104\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-calcular-el-producto-escalar-entre-dos-vectores\"><\/span>\u5982\u4f55\u8ba1\u7b97\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u70b9\u79ef<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u5728\u6570\u5b66\u4e2d\uff0c\u70b9\u79ef\u662f\u4e00\u79cd\u5411\u91cf\u8fd0\u7b97\uff0c\u5b83\u5c06\u4e24\u4e2a\u5411\u91cf\u76f8\u4e58\u5e76\u5c06\u5b83\u4eec\u8f6c\u6362\u4e3a\u5b9e\u6570\u3002\u56e0\u6b64\uff0c\u6709\u4e24\u79cd\u65b9\u6cd5\u53ef\u4ee5\u8ba1\u7b97\u4e24\u4e2a\u5411\u91cf\u7684\u70b9\u79ef\uff1a <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-105\"><\/div>\n<\/div>\n<div style=\"background-color:#FFCC8080;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px; border: 2px solid #FFB74D; border-radius:20px;\">\n<p style=\"text-align:left\">\u5982\u679c\u6211\u4eec\u77e5\u9053\u4e24\u4e2a\u5411\u91cf\u7684\u5750\u6807\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5c06 X \u548c Y \u5206\u91cf\u76f8\u4e58\u7136\u540e\u5c06\u7ed3\u679c\u76f8\u52a0\u6765\u627e\u5230\u5b83\u4eec\u7684\u70b9\u79ef\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5982\u679c\u6211\u4eec\u6709\u4e24\u4e2a\u5411\u91cf\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c06f6c1238ab7803750f830e16891f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (\\text{u}_x,\\text{u}_y) \\qquad \\vv{\\text{v}} = (\\text{v}_x,\\text{v}_y)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"216\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\">\u5b83\u4eec\u4e4b\u95f4\u7684\u6807\u91cf\u79ef\u4e3a\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c469402aae308ebcf911d56f71dadef2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\vv{\\text{u}} \\cdot \\vv{\\text{v}} = \\text{u}_x\\cdot \\text{v}_x + \\text{u}_y\\cdot \\text{v}_y\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"174\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<\/div>\n<p>\u4f8b\u5982\uff0c\u4ee5\u4e0b\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u70b9\u79ef\u4e3a\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-78adae8ec882bbd01c5bb2704f3ffa0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (1,2) \\qquad \\vv{\\text{v}} = (-1,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"194\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-756e86b9b5b7a9f5df2cacf728d82855_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{aligned} \\vv{\\text{u}} \\cdot \\vv{\\text{v}}&amp;=(1,2)\\cdot (-1,3) \\\\[1.5ex]&amp;=1\\cdot (-1) + 2 \\cdot 3 \\\\[1.5ex] &amp; = -1+6  \\\\[1.5ex] &amp; =\\bm{5} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"166\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u8fd9\u662f\u4e00\u79cd\u6c42\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u70b9\u79ef\u7684\u65b9\u6cd5\u3002\u4e0d\u8fc7\uff0c\u8fd8\u6709\u53e6\u4e00\u79cd\u65b9\u6cd5\uff1a <\/p>\n<div style=\"background-color:#FFCC8080;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px; border: 2px solid #FFB74D; border-radius:20px;\">\n<p style=\"text-align:left\">\u53e6\u4e00\u65b9\u9762\uff0c\u5982\u679c\u6211\u4eec\u77e5\u9053\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u6a21\u548c\u89d2\u5ea6\uff0c\u5219\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u6807\u91cf\u79ef\u53ef\u4ee5\u901a\u8fc7\u8ba1\u7b97\u5b83\u4eec\u7684\u6a21\u4e0e\u5b83\u4eec\u5f62\u6210\u7684\u89d2\u5ea6\u7684\u4f59\u5f26\u7684\u4e58\u79ef\u6765\u786e\u5b9a\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-46f454c7efa9e990e47b8fd3858fbade_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\vv{\\text{u}} \\cdot \\vv{\\text{v}} = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot \\cos(\\alpha )\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\">\u91d1\u5b50<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4501274336c637b37c6332eae5c6c229_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{u}} \\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"16\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u548c<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9a59cd4f2581db3318d38a2a77340a64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{v}} \\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"15\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u662f\u5411\u91cf\u7684\u6a21<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cac24ae79c1e4cbc459f01ed5e4f824e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p>\u548c<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-391ac2e3ba0b7f327ba5a0edc1ba162d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p>\u5206\u522b\u548c<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8f0b6b1a01f8fcc2f95be0364c090397_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p>\u4ed6\u4eec\u5f62\u6210\u7684\u89d2\u5ea6\u3002 <\/p>\n<\/div>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-106\"><\/div>\n<\/div>\n<p>\u56de\u60f3\u4e00\u4e0b\uff0c\u5411\u91cf\u7684\u5927\u5c0f\u662f\u5176\u5206\u91cf\u7684\u5e73\u65b9\u6839\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f63fa0a6f4110553705d4e3d6cf23692_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{u}} \\rvert = \\sqrt{ \\text{u}_x^2+\\text{u}_y^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"117\" style=\"vertical-align: -11px;\"><\/p>\n<\/p>\n<p>\u4f8b\u5982\uff0c\u6211\u4eec\u5c06\u6c42\u89e3\u4e24\u4e2a\u5411\u91cf\u7684\u6807\u91cf\u79ef\uff0c\u5176\u6a21\u6570\u548c\u5b83\u4eec\u4e4b\u95f4\u7684\u89d2\u5ea6\u4e3a\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-46c26555360d66fd213087ee2432e68e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{u}} \\rvert =3 \\qquad \\lvert \\vv{\\text{v}} \\rvert = 4 \\qquad \\alpha=60\u00ba\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"226\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a363625d93f01339437d5ad065050025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\vv{\\text{u}} \\cdot \\vv{\\text{v}} &amp; = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot \\cos(\\alpha ) \\\\[1.5ex] &amp;= 3 \\cdot 4 \\cdot \\cos(60\u00ba)\\\\[1.5ex] &amp; = 3 \\cdot 4 \\cdot 0,5 \\\\[1.5ex] &amp;= \\bm{6} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"168\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u53e6\u4e00\u65b9\u9762\uff0c\u70b9\u79ef\u4e5f\u79f0\u4e3a\u70b9\u79ef\u3001\u6807\u91cf\u79ef\u6216\u70b9\u79ef\u3002<\/p>\n<p><strong>\u6ce8\u610f\uff1a<\/strong>\u4e0d\u8981\u6df7\u6dc6\u70b9\u79ef\u548c\u53c9\u79ef\uff0c\u56e0\u4e3a\u867d\u7136\u5b83\u4eec\u7684\u540d\u79f0\u76f8\u4f3c\uff0c\u4f46\u5b83\u4eec\u662f\u5b8c\u5168\u4e0d\u540c\u7684\u6982\u5ff5\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"encontrar-el-angulo-entre-dos-vectores-utilizando-el-producto-escalar\"><\/span>\u4f7f\u7528\u70b9\u79ef\u6c42\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u89d2\u5ea6<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-109\"><\/div>\n<\/div>\n<p>\u4e00\u65e6\u6211\u4eec\u770b\u5230\u70b9\u79ef\u7684\u5b9a\u4e49\uff0c\u60a8\u53ef\u80fd\u60f3\u77e5\u9053\u4e24\u4e2a\u5411\u91cf\u76f8\u4e58\u7684\u76ee\u7684\u662f\u4ec0\u4e48\uff1f\u90a3\u4e48\uff0c\u70b9\u79ef\u7684\u5e94\u7528\u4e4b\u4e00\u5c31\u662f\u8ba1\u7b97\u4e24\u4e2a\u5411\u91cf\u5f62\u6210\u7684\u89d2\u5ea6\u3002 <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/angle-entre-deux-vecteurs-et-produit-scalaire.webp\" alt=\"\u4e24\u4e2a\u70b9\u79ef\u5411\u91cf\u4e4b\u95f4\u7684\u89d2\u5ea6\" class=\"wp-image-583\" width=\"172\" height=\"175\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p>\u901a\u8fc7\u6c42\u89e3\u70b9\u79ef\u516c\u5f0f\u7684\u4f59\u5f26\uff0c\u6211\u4eec\u5f97\u5230\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0534ed8d223cf93e44493b80ebfa83d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}\\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*} \\cos(\\alpha) =\\cfrac{\\vv{\\text{u}} \\cdot \\vv{\\text{v}}}{\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert}\\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u8ba9\u6211\u4eec\u901a\u8fc7\u4e00\u4e2a\u4f8b\u5b50\u770b\u770b\u8fd9\u662f\u5982\u4f55\u5b8c\u6210\u7684\uff1a<\/p>\n<ul>\n<li>\u6c42\u4ee5\u4e0b\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u89d2\u5ea6\uff1a<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d65d095d633d49431fcf3ee16757025e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (4,2) \\qquad \\vv{\\text{v}} = (-1,5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"194\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u9996\u5148\u6211\u4eec\u9700\u8981\u627e\u5230\u4e24\u4e2a\u5411\u91cf\u7684\u5927\u5c0f\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b8435f99e0a124c838bc163cb3ac1c67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{u}} \\rvert = \\sqrt{ 4^2+2^2}= \\sqrt{20}\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"170\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c8a9191c8cc87d5eeb9ba2adef3860f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{v}} \\rvert = \\sqrt{ (-1)^2+5^2}= \\sqrt{26}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"197\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u73b0\u5728\u6211\u4eec\u4f7f\u7528\u516c\u5f0f\u6765\u8ba1\u7b97\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u89d2\u5ea6\u7684\u4f59\u5f26\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-808047a2b37f1b5534038383a4d4b111_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\vv{\\text{u}} \\cdot \\vv{\\text{v}}}{\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert}=\\cfrac{ 4\\cdot (-1) + 2\\cdot 5}{\\sqrt{20}\\cdot \\sqrt{26}} = \\cfrac{6}{\\sqrt{520}} = 0,26\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"387\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p>\u6700\u540e\uff0c\u6211\u4eec\u7528\u8ba1\u7b97\u5668\u6c42\u4f59\u5f26\u7684\u5012\u6570\uff0c\u6c42\u51fa\u76f8\u5e94\u7684\u89d2\u5ea6\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36498fae23e4068e3d1c3735dafd2e64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos^{-1}(0,26) = \\bm{74,93\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"158\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\uff0c\u77e2\u91cf\u5f62\u6210\u7684\u89d2\u5ea6\u4e3a 74.93\u00b0\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedades-del-producto-escalar-de-dos-vectores\"><\/span>\u4e24\u4e2a\u5411\u91cf\u70b9\u79ef\u7684\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u70b9\u79ef\u5177\u6709\u4ee5\u4e0b\u7279\u70b9\uff1a<\/p>\n<ul>\n<li><strong>\u4ea4\u6362\u5f8b<\/strong>\uff1a\u5411\u91cf\u76f8\u4e58\u7684\u987a\u5e8f\u5e76\u4e0d\u91cd\u8981\u3002<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fcc9e9f7ec3121774701301a70313b86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}} \\cdot \\vv{\\text{v}} =\\vv{\\text{v}} \\cdot \\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"88\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li><strong>\u5206\u914d\u6027<\/strong>\uff1a\u70b9\u79ef\u5bf9\u4e8e\u5411\u91cf\u7684\u52a0\u6cd5\u548c\u51cf\u6cd5\u5177\u6709\u5206\u914d\u6027\uff1a<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0853c6a7d518e218e60c0d047bde152e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}( \\vv{\\text{v}}+ \\vv{\\text{w}} )=\\vv{\\text{u}} \\cdot \\vv{\\text{v}}+ \\vv{\\text{u}} \\cdot \\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"180\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-585d05c6a06b6c57b9ab0e640a371a9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}( \\vv{\\text{v}}- \\vv{\\text{w}} )=\\vv{\\text{u}} \\cdot \\vv{\\text{v}}- \\vv{\\text{u}} \\cdot \\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"180\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li><strong>\u5173\u8054\u6027<\/strong>\uff1a\u6211\u4eec\u53ef\u4ee5\u5728\u6267\u884c\u8fd0\u7b97\u4e4b\u524d\u6216\u4e4b\u540e\u5c06\u70b9\u79ef\u4e58\u4ee5\u4e00\u4e2a\u5e38\u6570\uff0c\u56e0\u4e3a\u7ed3\u679c\u662f\u7b49\u6548\u7684\uff1a<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1d8c1312b87c767bb4439f6c6c693dad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle k\\cdot (\\vv{\\text{u}} \\cdot \\vv{\\text{v}}) = (k\\cdot\\vv{\\text{u}}) \\cdot \\vv{\\text{v}} =\\vv{\\text{u}} \\cdot (k\\cdot\\vv{\\text{v}})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"252\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li>\u5982\u679c\u4e24\u4e2a\u5411\u91cf<strong>\u6b63\u4ea4<\/strong>\uff08\u6216\u5782\u76f4\uff09\uff0c\u5219\u5b83\u4eec\u7684\u70b9\u79ef\u4e3a\u96f6\u3002\u8fd9\u4e2a\u5c5e\u6027\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u8bc1\u660e\uff0c\u56e0\u4e3a\u4e24\u4e2a\u5782\u76f4\u5411\u91cf\u5f62\u6210 90\u00b0 \u7684\u89d2\u5ea6\uff0c\u5e76\u4e14 90\u00b0 \u7684\u4f59\u5f26\u7b49\u4e8e 0\uff1a<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-77e3b216b12f49a9dde99c0fbf626658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\vv{\\text{u}} \\cdot \\vv{\\text{v}} &amp; = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot \\cos(90\u00ba ) \\\\[1.5ex] &amp;=\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot 0 \\\\[1.5ex] &amp;= 0 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"91\" width=\"175\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li>\u76f8\u53cd\uff0c\u5982\u679c\u4e24\u4e2a\u5411\u91cf<strong>\u5e73\u884c<\/strong>\uff0c\u5219\u5b83\u4eec\u7684\u6807\u91cf\u79ef\u4e0e\u5b83\u4eec\u7684\u6a21\u7684\u4e58\u79ef\u76f8\u540c\u3002\u8fd9\u4e2a\u5c5e\u6027\u4e5f\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u9a8c\u8bc1\uff0c\u56e0\u4e3a\u4e24\u4e2a\u76f8\u540c\u65b9\u5411\u7684\u5411\u91cf\u5f62\u6210 0\u00b0 \u7684\u89d2\u5ea6\uff0c\u5176\u4f59\u5f26\u7b49\u4e8e 1\uff1a<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-777ec2544cac64aa065df21872989b4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\vv{\\text{u}} \\cdot \\vv{\\text{v}} &amp; = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot \\cos(0\u00ba) \\\\[1.5ex] &amp;=\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot 1 \\\\[1.5ex] &amp;= \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"96\" width=\"166\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li>\u6700\u540e\uff0c\u5411\u91cf\u672c\u8eab\u7684\u70b9\u79ef\u7b49\u4e8e\u5176\u5927\u5c0f\u7684\u5e73\u65b9\uff1a <\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0161de621030dcfe2a1d5fccc94048bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\vv{\\text{u}} \\cdot \\vv{\\text{u}} &amp; = \\lvert \\vv{\\text{u}} \\rvert ^2\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-productos-escalares-entre-dos-vectores\"><\/span>\u89e3\u51b3\u4e86\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u6807\u91cf\u79ef\u95ee\u9898<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e601<\/h3>\n<p>\u8ba1\u7b97\u4ee5\u4e0b\u4e24\u4e2a\u5411\u91cf\u5728\u5e73\u9762\u4e0a\u7684\u70b9\u79ef\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8c09eac43618120b3a2365fcd22278ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (4,-3) \\qquad \\vv{\\text{v}} = (5,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"194\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u8981\u8ba1\u7b97\u4e24\u4e2a\u5411\u91cf\u7684\u70b9\u79ef\uff0c\u6211\u4eec\u9700\u8981\u5c06\u5b83\u4eec\u7684 X \u5750\u6807\u548c Y \u5750\u6807\u76f8\u4e58\uff0c\u7136\u540e\u5c06\u7ed3\u679c\u76f8\u52a0\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f9cb0f372eee67ad149d5b2cff8d2f99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned}\\vv{\\text{u}} \\cdot \\vv{\\text{v}}  &amp; = (4,-3)\\cdot (5,2)  \\\\[1.5ex] &amp; = 4\\cdot 5 + (-3) \\cdot 2 \\\\[1.5ex] &amp; = 20-6\\\\[1.5ex] &amp; =\\bm{14} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"165\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e602<\/h3>\n<p>\u786e\u5b9a\u4e24\u4e2a\u5411\u91cf\u7684\u6807\u91cf\u79ef\uff0c\u5176\u6a21\u6570\u548c\u5b83\u4eec\u5f62\u6210\u7684\u89d2\u5ea6\u4e3a\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-55cc75d8aa625082e47ed6396449f550_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{u}} \\rvert =6 \\qquad \\lvert \\vv{\\text{v}} \\rvert = 3 \\qquad \\alpha=45\u00ba\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"225\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u7531\u4e8e\u6211\u4eec\u77e5\u9053\u5b83\u4eec\u7684\u6a21\u5757\u4ee5\u53ca\u5b83\u4eec\u4e4b\u95f4\u7684\u89d2\u5ea6\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u5e94\u7528\u70b9\u79ef\u516c\u5f0f\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-38999386ccbd92758c9968f025beff72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\vv{\\text{u}} \\cdot \\vv{\\text{v}} &amp; = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert \\cdot \\cos(\\alpha ) \\\\[1.5ex] &amp;= 6 \\cdot 3 \\cdot \\cos(45\u00ba)\\\\[1.5ex] &amp; = 6 \\cdot 3 \\cdot 0,71 \\\\[1.5ex] &amp;= \\bm{12,73} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"133\" width=\"168\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e603<\/h3>\n<p>\u4e0b\u9762\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u89d2\u5ea6\u662f\u591a\u5c11\uff1f <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-256770a8f550eb60828f1027831ec423_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}=(3,8) \\qquad  \\vv{\\text{v}} =(-4,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"194\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u9996\u5148\uff0c\u6211\u4eec\u9700\u8981\u8ba1\u7b97\u4e24\u4e2a\u5411\u91cf\u7684\u5927\u5c0f\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3a43677b59edab4031eae3cf64775e63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{u}} \\rvert = \\sqrt{ 3^2+8^2}= \\sqrt{73}\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"170\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-34ffe2db4622d64f22c5ec525dbc01ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{v}} \\rvert = \\sqrt{ (-4)^2+1^2}= \\sqrt{17}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"197\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u4f7f\u7528\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\u5411\u91cf\u5f62\u6210\u7684\u89d2\u5ea6\u7684\u4f59\u5f26\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eecb349d0dcd0362cfd6a67b639e4edb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\vv{\\text{u}} \\cdot \\vv{\\text{v}}}{\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert}=\\cfrac{ 3\\cdot (-4) + 8\\cdot 1}{\\sqrt{73}\\cdot \\sqrt{17}} = \\cfrac{-4}{\\sqrt{1241}} = -0,11\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"409\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u6211\u4eec\u901a\u8fc7\u8ba1\u7b97\u5668\u6c42\u4f59\u5f26\u7684\u5012\u6570\u6765\u627e\u5230\u76f8\u5e94\u7684\u89d2\u5ea6\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b3e2f0498a5af3652b484888d36d1f36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos^{-1}(-0,11) = \\bm{96,52\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"170\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e604<\/h3>\n<p>\u8003\u8651\u4ee5\u4e0b\u4e24\u4e2a\u5411\u91cf\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ae35d4962f496060eec23769832a6649_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}=(5,2) \\qquad \\vv{\\text{v}} =(-1,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"194\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u8ba1\u7b97\u5982\u4e0b\u8fd0\u7b97\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-256d0ec6c6c20b5914f3ab0716119a97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 \\bigl(\\vv{\\text{u}} \\cdot\\vv{\\text{v}}\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"55\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u6211\u4eec\u9996\u5148\u9700\u8981\u6c42\u89e3\u62ec\u53f7\u5185\u7684\u70b9\u79ef\uff0c\u7136\u540e\u4e0e\u62ec\u53f7\u5916\u7684\u70b9\u79ef\u76f8\u4e58\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-256d0ec6c6c20b5914f3ab0716119a97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 \\bigl(\\vv{\\text{u}} \\cdot\\vv{\\text{v}}\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"55\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8652a87d0713db4774dae56ca4328e49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 \\bigl((5,2) \\cdot (-1,6) \\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"129\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c6f9d3b4f48b49bdfdfc69a19dd13903_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 \\bigl(5 \\cdot (-1) + 2 \\cdot 6 \\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"133\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-afe85bb11fcdcc61ae43a2d7f6b7603c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 \\bigl(-5 + 12 \\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"85\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-64de0e6750bcabc711c6d73ec0d0d869_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4 \\cdot 7\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-32d67549905fec64c67a14e7a8b694e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{28}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e605<\/h3>\n<p>\u7ed9\u5b9a\u4ee5\u4e0b\u4e09\u4e2a\u4e8c\u7ef4\u5411\u91cf\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-202792a7dea6027ec366cffa5ff30e22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}=(-2,6) \\qquad \\vv{\\text{v}} =(4,-3)\\qquad \\vv{\\text{w}} =(-1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"333\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u8ba1\u7b97\u5982\u4e0b\u8fd0\u7b97\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8413dfd8859a9b5a549dbfd002683439_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}} \\cdot \\bigl( 5 \\vv{\\text{u}}- 2 \\vv{\\text{v}}\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"98\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u9996\u5148\uff0c\u6211\u4eec\u5c06\u5411\u91cf\u4e58\u4ee5\u62ec\u53f7\u4e2d\u7684\u6807\u91cf\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8413dfd8859a9b5a549dbfd002683439_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}} \\cdot \\bigl( 5 \\vv{\\text{u}}- 2 \\vv{\\text{v}}\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"98\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dc363692e0558fc697dee94018108f2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1,2) \\cdot \\bigl( 5 (-2,6)- 2(4,-3)\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"225\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c51b85108b7dc83bc81a75702db939bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1,2) \\cdot \\bigl( (-10,30)- (8,-6)\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"225\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u73b0\u5728\u6211\u4eec\u8fdb\u884c\u5411\u91cf\u51cf\u6cd5\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ca704e156d875e9ca62b62452c77cc3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1,2) \\cdot  (-10 -8,30-(-6))\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"224\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-28e80b50176016afcc5901693ef5426d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1,2) \\cdot  (-18,36)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"135\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u6211\u4eec\u6c42\u89e3\u6807\u91cf\u79ef\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-498a8066459db0849a08866445623dbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(-1)\\cdot (-18) + 2 \\cdot 36\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"155\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f3c885a9e81d347e3914a736b32bb382_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"18 + 72\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"55\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-56a66a32d168d33967dda62795778226_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{90}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e606<\/h3>\n<p>\u8ba1\u7b97\u503c<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p>\u4f7f\u5f97\u4ee5\u4e0b\u5411\u91cf\u4e92\u76f8\u5782\u76f4\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1b1c6281d6e6570fd94edcd837293e2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vv{\\text{u}}=(-2,-3) \\qquad  \\vv{\\text{v}} =(k,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"208\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u4e24\u4e2a\u5782\u76f4\u5411\u91cf\u5f62\u6210 90\u00b0 \u89d2\u3002\u56e0\u6b64\u89d2\u5ea6\u7684\u4f59\u5f26\u5fc5\u987b\u4e3a\u96f6\uff0c\u56e0\u4e3a cos(90\u00b0)=0\u3002\u7136\u800c\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e83a6b694c8dfa0975854f1bffec44de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(90\u00ba) =\\cfrac{\\vv{\\text{u}} \\cdot \\vv{\\text{v}}}{\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"133\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3ef11c2ecbf7bc8dff4217a761960387_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0=\\cfrac{\\vv{\\text{u}} \\cdot \\vv{\\text{v}}}{\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"86\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u5206\u6570\u7684\u5206\u6bcd\u9664\u4ee5\u65b9\u7a0b\u7684\u6574\u4e2a\u53f3\u4fa7\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5728\u53e6\u4e00\u4fa7\u76f8\u4e58\u6765\u4f20\u9012\u5b83\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-797dd0ce47130f959c984510894f08b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 \\cdot \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert  =\\vv{\\text{u}} \\cdot \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b58e54d3d5fa6e123ca5e27a27d77ad1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0  =\\vv{\\text{u}} \\cdot \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"64\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u73b0\u5728\u6c42\u89e3\u6807\u91cf\u79ef\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-67bb2b17e1eb4d090327e96f5f3a8bcf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =(-2,-3) \\cdot (k,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"152\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-77ccd428939d3270f6feeef3ca9681e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =-2 \\cdot k + (-3)\\cdot 6\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4dacdcd81cad7dcb737b48de38e3b4a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =-2 k -18\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"105\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u6211\u4eec\u6f84\u6e05\u672a\u77e5\u7684\u4e8b\u60c5\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-35a0330d3e83b419577d9448ff01008f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2k =-18\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"74\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6abbb3c976dc4f8d66093a3fb3a40cca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle k =\\cfrac{-18}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"75\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-67556bf37fde34c4177ff3f3c037f95c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{k =-9}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e607<\/h3>\n<p>\u8ba1\u7b97\u89d2\u5ea6<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-28575fb8fa361427b255d8744e982cf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha , \\beta\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"30\" style=\"vertical-align: -4px;\"><\/p>\n<p>\u548c<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4de02fc502ed5dbd15f371728ea270a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\gamma\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\"><\/p>\n<p>\u5f62\u6210\u4ee5\u4e0b\u4e09\u89d2\u5f62\u7684\u8fb9\uff1a <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-angle-resolu-entre-vecteurs-produit-scalaire.webp\" alt=\"\u4e24\u4e2a\u5411\u91cf\u7684\u6807\u91cf\u79ef\u7684\u7ec3\u4e60\u548c\u95ee\u9898\u7684\u9010\u6b65\u89e3\u51b3\" class=\"wp-image-560\" width=\"290\" height=\"226\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u7ec4\u6210\u4e09\u89d2\u5f62\u7684\u9876\u70b9\u662f\u4ee5\u4e0b\u70b9\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-75a4919fae29190e3effdeedcec8eb6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(2,1) \\qquad B(4,4) \\qquad C(6,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"230\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u8981\u8ba1\u7b97\u4e09\u89d2\u5f62\u7684\u5185\u89d2\uff0c\u6211\u4eec\u53ef\u4ee5\u8ba1\u7b97\u5176\u6bcf\u6761\u8fb9\u7684\u5411\u91cf\uff0c\u7136\u540e\u4f7f\u7528\u70b9\u79ef\u516c\u5f0f\u627e\u5230\u5b83\u4eec\u5f62\u6210\u7684\u89d2\u5ea6\u3002<\/p>\n<p class=\"has-text-align-left\">\u4f8b\u5982\uff0c\u6c42\u89d2\u5ea6<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8f0b6b1a01f8fcc2f95be0364c090397_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p>\u6211\u4eec\u8ba1\u7b97\u5176\u8fb9\u7684\u5411\u91cf\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6a9da14fa9cc4e50b06bdfa76801b083_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB} = B - A = (4,4)-(2,1)= (2,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"287\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8e4e2e72bee87bba3e7657a53935e660_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AC} = C - A = (6,2)-(2,1)= (4,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"286\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u4f7f\u7528\u70b9\u79ef\u516c\u5f0f\u6c42\u51fa\u4e24\u4e2a\u5411\u91cf\u5f62\u6210\u7684\u89d2\u5ea6\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7b6aad49b300d421fc3bb486f051294c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{AB} \\rvert = \\sqrt{2^2+3^2} = \\sqrt{13}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"185\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f6657897e68d6b68f79277c89abe6868_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{AC} \\rvert = \\sqrt{4^2+1^2} = \\sqrt{17}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"185\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a966db5753cbb53c424c0f962fb27102_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\vv{AB} \\cdot \\vv{AC}}{\\lvert \\vv{AB} \\rvert \\cdot \\lvert \\vv{AC} \\rvert}=\\cfrac{ 2\\cdot 4 + 3\\cdot 1}{\\sqrt{13}\\cdot \\sqrt{17}} = \\cfrac{11}{\\sqrt{221}} =0,74\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"396\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9fac783dc0113263dfb5c31b58231fae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\alpha = 42,27\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"79\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u73b0\u5728\u6211\u4eec\u91cd\u590d\u76f8\u540c\u7684\u8fc7\u7a0b\u6765\u786e\u5b9a\u89d2\u5ea6<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ea160d5901518098e691e051e6efa4a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\beta:\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"20\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-398c0b2dc840abfc63700a084e9e2956_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{BC} = C - B = (6,2)-(4,4)= (2,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"302\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b7825d9e3b0ceee57e7ecd470e52a242_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{BC} \\rvert = \\sqrt{2^2+(-2)^2} = \\sqrt{8}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"207\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e0e73fd58d6a5b487af9f971fdcdc97f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\beta) =\\cfrac{\\vv{AB} \\cdot \\vv{BC}}{\\lvert \\vv{AB} \\rvert \\cdot \\lvert \\vv{BC} \\rvert}=\\cfrac{ 2\\cdot 2 + 3\\cdot (-2)}{\\sqrt{13}\\cdot \\sqrt{8}} = \\cfrac{-2}{\\sqrt{104}} =-0,20\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"437\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1b2f148d28b9679b8267886497e16518_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\beta = 101,31\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"86\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u4e3a\u4e86\u627e\u5230\u6700\u540e\u4e00\u4e2a\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u91cd\u590d\u76f8\u540c\u7684\u8fc7\u7a0b\u3002\u7136\u800c\uff0c\u4e09\u89d2\u5f62\u4e2d\u7684\u6240\u6709\u89d2\u4e4b\u548c\u5fc5\u987b\u4e3a 180 \u5ea6\uff0c\u56e0\u6b64\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-662cae07e8d96d1164dad2b0358302fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\gamma = 180 -42,27-101,31 = \\bm{36,42\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"266\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u5728\u6b64\u9875\u9762\u4e0a\uff0c\u60a8\u5c06\u4e86\u89e3\u5b83\u662f\u4ec0\u4e48\u4ee5\u53ca\u5982\u4f55\u8ba1\u7b97\u4e24\u4e2a\u5411\u91cf\u7684\u70b9\u79ef\u3002\u60a8\u8fd8\u5c06\u5b66\u4e60\u5982\u4f55\u4f7f\u7528\u70b9\u79ef\u6c42\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u89d2\u5ea6\uff0c\u6b64\u5916\u8fd8\u5b66\u4e60 &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/cn\/%e8%ae%a1%e7%ae%97%e4%b8%a4%e4%b8%aa%e5%90%91%e9%87%8f%e4%b9%8b%e9%97%b4%e7%9a%84%e6%a0%87%e9%87%8f%e7%a7%af-%e5%b7%b2%e8%a7%a3%e5%86%b3%e7%9a%84%e7%a4%ba%e4%be%8b%e7%bb%83%e4%b9%a0\/\"> <span class=\"screen-reader-text\">\u8ba1\u7b97\u4e24\u4e2a\u5411\u91cf\u7684\u70b9\u79ef<\/span> \u67e5\u770b\u5168\u6587 &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[23],"tags":[],"class_list":["post-211","post","type-post","status-publish","format-standard","hentry","category-23"],"yoast_head":"<!-- This site is optimized 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