{"id":249,"date":"2023-07-10T11:53:09","date_gmt":"2023-07-10T11:53:09","guid":{"rendered":"https:\/\/mathority.org\/cn\/%e7%a9%ba%e9%97%b4%e4%b8%ad%e4%b8%a4%e4%b8%aa%e5%b9%b3%e9%9d%a2%e4%b9%8b%e9%97%b4%e7%9a%84%e8%a7%92%e5%ba%a6%e5%85%ac%e5%bc%8f-r3\/"},"modified":"2023-07-10T11:53:09","modified_gmt":"2023-07-10T11:53:09","slug":"%e7%a9%ba%e9%97%b4%e4%b8%ad%e4%b8%a4%e4%b8%aa%e5%b9%b3%e9%9d%a2%e4%b9%8b%e9%97%b4%e7%9a%84%e8%a7%92%e5%ba%a6%e5%85%ac%e5%bc%8f-r3","status":"publish","type":"post","link":"https:\/\/mathority.org\/cn\/%e7%a9%ba%e9%97%b4%e4%b8%ad%e4%b8%a4%e4%b8%aa%e5%b9%b3%e9%9d%a2%e4%b9%8b%e9%97%b4%e7%9a%84%e8%a7%92%e5%ba%a6%e5%85%ac%e5%bc%8f-r3\/","title":{"rendered":"\u7a7a\u95f4\u4e2d\u4e24\u4e2a\u5e73\u9762\u4e4b\u95f4\u7684\u89d2\u5ea6\uff08\u516c\u5f0f\uff09"},"content":{"rendered":"<p>\u5728\u6b64\u9875\u9762\u4e0a\uff0c\u60a8\u5c06\u4e86\u89e3\u5982\u4f55\u8ba1\u7b97\u7a7a\u95f4\u4e2d\u4e24\u4e2a\u5e73\u9762\u5f62\u6210\u7684\u89d2\u5ea6\uff08\u516c\u5f0f\uff09\u3002\u6b64\u5916\uff0c\u60a8\u5c06\u80fd\u591f\u770b\u5230\u793a\u4f8b\u5e76\u901a\u8fc7\u5df2\u89e3\u51b3\u7684\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=\"formula-del-angulo-entre-dos-planos\"><\/span>\u4e24\u4e2a\u5e73\u9762\u4e4b\u95f4\u7684\u89d2\u5ea6\u516c\u5f0f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>\u4e24\u4e2a\u5e73\u9762\u4e4b\u95f4\u7684\u89d2\u5ea6\u7b49\u4e8e\u6240\u8ff0\u5e73\u9762\u7684\u6cd5\u5411\u91cf\u5f62\u6210\u7684\u89d2\u5ea6\u3002\u56e0\u6b64\uff0c\u4e3a\u4e86\u627e\u5230\u4e24\u4e2a\u5e73\u9762\u4e4b\u95f4\u7684\u89d2\u5ea6\uff0c\u9700\u8981\u8ba1\u7b97\u5b83\u4eec\u7684\u6cd5\u5411\u91cf\u5f62\u6210\u7684\u89d2\u5ea6\uff0c\u56e0\u4e3a\u5b83\u4eec\u662f\u76f8\u7b49\u7684\u3002<\/strong> <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-105\"><\/div>\n<\/div>\n<p>\u56e0\u6b64\uff0c\u4e00\u65e6\u6211\u4eec\u786e\u5207\u5730\u77e5\u9053\u4e86\u4e24\u4e2a\u5e73\u9762\u4e4b\u95f4\u7684\u89d2\u5ea6\u662f\u591a\u5c11\uff0c\u6211\u4eec\u5c31\u770b\u4e00\u4e0b\u8ba1\u7b97\u7a7a\u95f4\u4e2d\u4e24\u4e2a\u5e73\u9762\u4e4b\u95f4\u7684\u89d2\u5ea6\u7684\u516c\u5f0f\uff08\u5728 R3 \u4e2d\uff09\uff0c\u8be5\u516c\u5f0f\u662f\u4ece<a href=\"https:\/\/mathority.org\/cn\/\u5982\u4f55\u8ba1\u7b97\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u89d2\u5ea6\u793a\u4f8b\u5df2\u89e3\u51b3\u7ec3\u4e60\/\">\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u89d2\u5ea6\u516c\u5f0f<\/a>\u63a8\u5bfc\u51fa\u6765\u7684\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\">\u7ed9\u5b9a\u4e24\u4e2a\u4e0d\u540c\u5e73\u9762\u7684\u4e00\u822c\uff08\u6216\u9690\u5f0f\uff09\u65b9\u7a0b\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-dfa3d7e6f1ece8353327be7c9227d75b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_1 : \\ A_1x+B_1y+C_1z+D_1=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"249\" 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-2c3966346685421fe3e535cf57a5491d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_2 : \\ A_2x+B_2y+C_2z+D_2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"249\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\">\u6bcf\u4e2a\u5e73\u9762\u7684\u6cd5\u5411\u91cf\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-eb0ca06882e0d61d6f8134368946ef29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_1=(A_1,B_1,C_1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" 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-22fba6a063a544bdf257e64d8d139238_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_2=(A_2,B_2,C_2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\">\u8fd9\u4e24\u4e2a\u5e73\u9762\u5f62\u6210\u7684\u89d2\u5ea6\u662f\u901a\u8fc7\u4f7f\u7528\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\u5b83\u4eec\u7684\u6cd5\u5411\u91cf\u5f62\u6210\u7684\u89d2\u5ea6\u6765\u786e\u5b9a\u7684\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-48fc901ce118ca0f0daecdf37b011101_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"145\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<\/div>\n<p>\u56e0\u6b64\uff0c\u8981\u786e\u5b9a\u4e24\u4e2a\u5e73\u9762\u4e4b\u95f4\u7684\u89d2\u5ea6\uff0c\u5fc5\u987b\u638c\u63e1 <a href=\"https:\/\/mathority.org\/cn\/\u8ba1\u7b97\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u6807\u91cf\u79ef-\u5df2\u89e3\u51b3\u7684\u793a\u4f8b\u7ec3\u4e60\/\">\u4e24\u4e2a\u5411\u91cf\u70b9\u79ef<\/a>\u7684\u8ba1\u7b97\u3002\u5982\u679c\u60a8\u4e0d\u8bb0\u5f97\u5b83\u662f\u5982\u4f55\u5b8c\u6210\u7684\uff0c\u5728\u94fe\u63a5\u4e2d\u60a8\u5c06\u627e\u5230\u89e3\u51b3\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u70b9\u79ef\u7684\u6b65\u9aa4\u3002\u6b64\u5916\uff0c\u60a8\u5c06\u80fd\u591f\u770b\u5230\u9010\u6b65\u89e3\u51b3\u7684\u793a\u4f8b\u548c\u7ec3\u4e60\u3002<\/p>\n<p>\u53e6\u4e00\u65b9\u9762\uff0c\u5f53\u4e24\u4e2a\u5e73\u9762\u5782\u76f4\u6216\u5e73\u884c\u65f6\uff0c\u5219\u4e0d\u5fc5\u5e94\u7528\u8be5\u516c\u5f0f\uff0c\u56e0\u4e3a\u53ef\u4ee5\u76f4\u63a5\u786e\u5b9a\u4e24\u4e2a\u5e73\u9762\u4e4b\u95f4\u7684\u89d2\u5ea6\uff1a<\/p>\n<ul>\n<li>\u4e24\u4e2a<strong>\u5e73\u884c\u5e73\u9762<\/strong>\u4e4b\u95f4\u7684\u89d2\u5ea6\u4e3a 0\u00b0\uff0c\u56e0\u4e3a\u5b83\u4eec\u7684\u6cd5\u5411\u91cf\u5177\u6709\u76f8\u540c\u7684\u65b9\u5411\u3002<\/li>\n<li>\u4e24\u4e2a<strong>\u5782\u76f4\u5e73\u9762<\/strong>\u4e4b\u95f4\u7684\u89d2\u5ea6\u4e3a 90\u00b0\uff0c\u56e0\u4e3a\u5b83\u4eec\u7684\u6cd5\u5411\u91cf\u4e5f\u5f7c\u6b64\u5782\u76f4\uff08\u6216\u6b63\u4ea4\uff09\uff0c\u56e0\u6b64\u5f62\u6210\u76f4\u89d2\u3002 <\/li>\n<\/ul>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-como-calcular-el-angulo-entre-dos-planos\"><\/span>\u8ba1\u7b97\u4e24\u4e2a\u5e73\u9762\u4e4b\u95f4\u7684\u89d2\u5ea6\u7684\u793a\u4f8b<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-106\"><\/div>\n<\/div>\n<p>\u8fd9\u662f\u4e00\u4e2a\u5177\u4f53\u793a\u4f8b\uff0c\u60a8\u53ef\u4ee5\u4e86\u89e3\u5982\u4f55\u786e\u5b9a\u4e24\u4e2a\u4e0d\u540c\u5e73\u9762\u4e4b\u95f4\u7684\u89d2\u5ea6\uff1a<\/p>\n<ul>\n<li>\u8ba1\u7b97\u4ee5\u4e0b\u4e24\u4e2a\u5e73\u9762\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-60b094e253552cbfa84175e60ef18801_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_1 : \\ 3x-5y+z+4=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"191\" 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-364b525ebfd0b5ef85128562d1641cb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_2 : \\ 4x+2y+3z-1=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"200\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p>\u6211\u4eec\u9700\u8981\u505a\u7684\u7b2c\u4e00\u4ef6\u4e8b\u662f\u627e\u5230\u6bcf\u4e2a\u5e73\u9762\u7684\u6cd5\u5411\u91cf\u3002\u56e0\u6b64\uff0c\u5782\u76f4\u4e8e\u5e73\u9762\u7684\u5411\u91cf\u7684\u5750\u6807 X\u3001Y\u3001Z \u5206\u522b\u4e0e\u5176\u901a\u5f0f\uff08\u6216\u9690\u5f0f\uff09\u65b9\u7a0b\u7684\u7cfb\u6570 A\u3001B \u548c C \u91cd\u5408\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-17c8988d3ee87dd9e708e46aadbb9086_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_1 = (3,-5,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" 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-5783f55ed327128d1da574f38c8336e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_2 = (4,2,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u4e00\u65e6\u6211\u4eec\u77e5\u9053\u4e86\u6bcf\u4e2a\u5e73\u9762\u7684\u6cd5\u5411\u91cf\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u7528\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\u5b83\u4eec\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-48fc901ce118ca0f0daecdf37b011101_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"145\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\uff0c\u6211\u4eec\u5fc5\u987b\u627e\u5230\u6bcf\u4e2a\u6cd5\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-96718e6e72f0f3045ee39364f636b419_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{3^2+(-5)^2+1^2}= \\sqrt{9+25+1} = \\sqrt{35}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"311\" 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-e3558cfa79b7113b469df5b36e00f1ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{4^2+2^2+3^2}= \\sqrt{16+4+9} = \\sqrt{29}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"280\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p>\u73b0\u5728\u6211\u4eec\u5c06\u6bcf\u4e2a\u672a\u77e5\u6570\u7684\u503c\u4ee3\u5165\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-5cdc9e54b22394cf29f35c6c26ee1308_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}=\\cfrac{\\lvert (3,-5,1) \\cdot (4,2,3)\\rvert}{\\sqrt{35} \\cdot \\sqrt{29} }\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"318\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p>\u6211\u4eec\u901a\u8fc7\u6c42\u89e3\u4e24\u4e2a\u5411\u91cf\u7684\u70b9\u79ef\u6765\u8ba1\u7b97\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-b1923cb6ac441ba0e980bc984bc4804b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert 3\\cdot 4 + (-5)\\cdot 2 +1 \\cdot 3 \\rvert}{\\sqrt{35}\\cdot \\sqrt{29} }=\\cfrac{\\lvert 12-10+3 \\rvert}{\\sqrt{1015}}= \\cfrac{5}{31,86}=0,16\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"497\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p>\u6700\u540e\uff0c\u6211\u4eec\u901a\u8fc7\u4f7f\u7528\u8ba1\u7b97\u5668\u8ba1\u7b97\u4f59\u5f26\u7684\u5012\u6570\u6765\u786e\u5b9a\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-57297e96c6a14eed6ff87abfe7699df5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\alpha = \\cos^{-1}(0,16)=\\bm{80,97\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"193\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-del-angulo-entre-dos-planos\"><\/span>\u89e3\u51b3\u4e86\u4e24\u4e2a\u5e73\u9762\u4e4b\u95f4\u7684\u89d2\u5ea6\u95ee\u9898<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<h3 class=\"wp-block-heading\">\u7ec3\u4e601<\/h3>\n<p>\u6c42\u4ee5\u4e0b\u4e24\u4e2a\u5e73\u9762\u4e4b\u95f4\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-ac5eb1f4c801eae9ea93342c56e7aa60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_1 : \\ x+2z-5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"151\" style=\"vertical-align: -3px;\"><\/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-f7b262c6e50b562858b1c5043548ba75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_2 : \\ 3x+y-4z+7=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"191\" style=\"vertical-align: -4px;\"><\/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\u9700\u8981\u505a\u7684\u7b2c\u4e00\u4ef6\u4e8b\u662f\u627e\u5230\u6bcf\u4e2a\u5e73\u9762\u7684\u6cd5\u5411\u91cf\u3002\u56e0\u6b64\uff0c\u5782\u76f4\u4e8e\u5e73\u9762\u7684\u5411\u91cf\u7684\u5750\u6807 X\u3001Y\u3001Z \u5206\u522b\u76f8\u5f53\u4e8e\u5176\u901a\u5f0f\uff08\u6216\u9690\u5f0f\uff09\u65b9\u7a0b\u7684\u7cfb\u6570 A\u3001B \u548c C\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-ef6a043fab595442ed8b17da7798fc34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_1 = (1,0,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" 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-158dadd8329a3954114f9d99160e9800_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_2 = (3,1,-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4e00\u65e6\u6211\u4eec\u77e5\u9053\u4e86\u6bcf\u4e2a\u5e73\u9762\u7684\u6cd5\u5411\u91cf\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u7528\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\u5b83\u4eec\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-48fc901ce118ca0f0daecdf37b011101_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"145\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\uff0c\u6211\u4eec\u5fc5\u987b\u627e\u5230\u6bcf\u4e2a\u6cd5\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-303393672cb4ca6c891bfcf49d3e30b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{1^2+0^2+2^2}= \\sqrt{1+4} = \\sqrt{5}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"232\" style=\"vertical-align: -3px;\"><\/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-ede42afce2c7b11dd96617df978ee401_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{3^2+1^2+(-4)^2}= \\sqrt{9+1+16} = \\sqrt{26}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"311\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u5c06\u6bcf\u4e2a\u672a\u77e5\u6570\u7684\u503c\u4ee3\u5165\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-90594e9bdf100d52e0221a86d0878d16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}=\\cfrac{\\lvert (1,0,2) \\cdot (3,1,-4)\\rvert}{\\sqrt{5} \\cdot \\sqrt{26} }\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"318\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u8ba1\u7b97\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-fb411278d0746c45c8ef0490e51862b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert 1\\cdot 3 + 0\\cdot 1 +2 \\cdot (-4) \\rvert}{\\sqrt{5}\\cdot \\sqrt{26} }=\\cfrac{\\lvert 3-8 \\rvert}{\\sqrt{130}}= \\cfrac{5}{11,4}=0,44\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"440\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u6211\u4eec\u901a\u8fc7\u8ba1\u7b97\u5668\u53cd\u8f6c\u4f59\u5f26\u6765\u627e\u5230\u4e24\u4e2a\u5e73\u9762\u4e4b\u95f4\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-aa9117c27a8c072f275a8cb5fac99528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\alpha = \\cos^{-1}(0,44)=\\bm{63,99\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"193\" 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\u4e602<\/h3>\n<p>\u4e0b\u9762\u4e24\u4e2a\u5e73\u9762\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-7566850c5f664f60ba826bb1ffbf7a3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_1 : \\ 3x-2y+5z=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"170\" 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-ead661dfbeded4be15602ccda3f2864c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_2 : \\ 6x+3y-z-2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"191\" style=\"vertical-align: -4px;\"><\/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\u9700\u8981\u505a\u7684\u7b2c\u4e00\u4ef6\u4e8b\u662f\u627e\u5230\u6bcf\u4e2a\u5e73\u9762\u7684\u6cd5\u5411\u91cf\u3002\u56e0\u6b64\uff0c\u5782\u76f4\u4e8e\u5e73\u9762\u7684\u5411\u91cf\u7684 X\u3001Y\u3001Z \u5750\u6807\u5206\u522b\u7b49\u4e8e\u5176\u901a\u5f0f\uff08\u6216\u9690\u5f0f\uff09\u65b9\u7a0b\u7684\u53c2\u6570 A\u3001B \u548c C\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-54788d8fdba6722dc87ae3c06c350400_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_1 = (3,-2,5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" 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-f6dee74c50bac2e8859c3f4a3c9742f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_2 = (6,3,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4e00\u65e6\u6211\u4eec\u77e5\u9053\u4e86\u6bcf\u4e2a\u5e73\u9762\u7684\u6cd5\u5411\u91cf\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u7528\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\u5b83\u4eec\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-48fc901ce118ca0f0daecdf37b011101_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"145\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\uff0c\u6211\u4eec\u5fc5\u987b\u627e\u5230\u6bcf\u4e2a\u6cd5\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-06daed2734e23937ffc63d52314656ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{3^2+(-2)^2+5^2}= \\sqrt{9+4+25} = \\sqrt{38}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"311\" 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-0f0bafe4966148f6e199dbd2bdc7ede1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{6^2+3^2+(-1)^2}= \\sqrt{36+9+1} = \\sqrt{46}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"311\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u5c06\u6bcf\u4e2a\u53d8\u91cf\u7684\u503c\u4ee3\u5165\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-3d635cbe5718a970df439157ca6346cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}=\\cfrac{\\lvert (3,-2,5) \\cdot (6,3,-1)\\rvert}{\\sqrt{38} \\cdot \\sqrt{46} }\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"332\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u8ba1\u7b97\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-a24e1ac8ad5f23b1d0734a564fa92a90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert 3\\cdot 6 + (-2)\\cdot 3 +5 \\cdot (-1) \\rvert}{\\sqrt{38}\\cdot \\sqrt{46} }=\\cfrac{\\lvert 18-6-5 \\rvert}{\\sqrt{1748}}= \\cfrac{7}{41,81}=0,17\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"516\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u6211\u4eec\u901a\u8fc7\u4f7f\u7528\u8ba1\u7b97\u5668\u53cd\u8f6c\u4f59\u5f26\u6765\u786e\u5b9a\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-d8fddc88af864b95ff37c56b126af7a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\alpha = \\cos^{-1}(0,17)=\\bm{80,36\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"193\" 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\u4e603<\/h3>\n<p>\u8ba1\u7b97\u53c2\u6570\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\u4e24\u4e2a\u5e73\u9762\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-faed0ccf5807a84f20057b0128bbcee5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_1 : \\ x+2y-3z+1=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"191\" 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-9872d7160fb07e1a3ca43e18e24b3435_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_2 : \\ -2x+5y+kz+4=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"215\" style=\"vertical-align: -4px;\"><\/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\u8981\u8ba1\u7b97\u5e73\u9762\u4e4b\u95f4\u7684\u89d2\u5ea6\uff0c\u60a8\u59cb\u7ec8\u5fc5\u987b\u627e\u5230\u6bcf\u4e2a\u5e73\u9762\u7684\u6cd5\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-add7176bfadc88b0c27dd09e33b340cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_1 = (1,2,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" 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-a3f04e685fb9ce2300340f149aa93059_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_2 = (-2,5,k)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"112\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4e24\u4e2a\u5782\u76f4\u5e73\u9762\u5f62\u6210 90\u00b0 \u7684\u89d2\u5ea6\uff0c\u56e0\u6b64\u5b83\u4eec\u7684\u6cd5\u5411\u91cf\u4e5f\u5c06\u662f 90\u00b0\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u786e\u5b9a\u672a\u77e5\u6570\u7684\u503c\u3002<\/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>\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u89d2\u5ea6\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-48fc901ce118ca0f0daecdf37b011101_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"145\" 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-a35dde3971efe5cdf36340847dcb02e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(90\u00ba) =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"151\" 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-eb4652afcc23693b300da306909293ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"104\" 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-f94a174ae36525a6e6ef055e53bb154a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 \\cdot \\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert =\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"172\" 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-e4cffd0cb6eef3c14b8924706cca4a43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =\\vv{n}_1 \\cdot \\vv{n}_2\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"82\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u73b0\u5728\u6211\u4eec\u6c42\u89e3\u4e24\u4e2a\u6cd5\u5411\u91cf\u4e4b\u95f4\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-3818384c903fe4f03fd785f0fbfb0197_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =(1,2,-3) \\cdot (-2,5,k)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" 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-a0317a3a3ce7f7a3b19d199026305b93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =1 \\cdot (-2) + 2\\cdot 5 +(-3)\\cdot k\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"223\" 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-c9a5497303548b34fd3bba83bdc588b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =-2 +10-3k\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"134\" 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-3f68c05212b2541bc6d1142287eebc18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0 =8-3k\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"81\" 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-5c13e299367327b12f0f781a10c47f12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 3k=8\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"51\" 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-b126ac6d6422d2ae3c536589c2461e2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{k =}\\mathbf{\\cfrac{8}{3}}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"41\" style=\"vertical-align: -12px;\"><\/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\u5982\u4f55\u8ba1\u7b97\u7a7a\u95f4\u4e2d\u4e24\u4e2a\u5e73\u9762\u5f62\u6210\u7684\u89d2\u5ea6\uff08\u516c\u5f0f\uff09\u3002\u6b64\u5916\uff0c\u60a8\u5c06\u80fd\u591f\u770b\u5230\u793a\u4f8b\u5e76\u901a\u8fc7\u5df2\u89e3\u51b3\u7684\u7ec3\u4e60\u8fdb\u884c\u7ec3\u4e60 &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/cn\/%e7%a9%ba%e9%97%b4%e4%b8%ad%e4%b8%a4%e4%b8%aa%e5%b9%b3%e9%9d%a2%e4%b9%8b%e9%97%b4%e7%9a%84%e8%a7%92%e5%ba%a6%e5%85%ac%e5%bc%8f-r3\/\"> <span class=\"screen-reader-text\">\u7a7a\u95f4\u4e2d\u4e24\u4e2a\u5e73\u9762\u4e4b\u95f4\u7684\u89d2\u5ea6\uff08\u516c\u5f0f\uff09<\/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":[15],"tags":[],"class_list":["post-249","post","type-post","status-publish","format-standard","hentry","category-15"],"yoast_head":"<!-- This site is optimized 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