{"id":244,"date":"2023-07-10T14:01:40","date_gmt":"2023-07-10T14:01:40","guid":{"rendered":"https:\/\/mathority.org\/cn\/%e5%b9%b3%e9%9d%a2%e7%9a%84%e9%9a%90%e5%bc%8f%e4%b8%80%e8%88%ac%e6%96%b9%e7%a8%8b%e6%88%96%e7%ac%9b%e5%8d%a1%e5%b0%94%e6%96%b9%e7%a8%8b\/"},"modified":"2023-07-10T14:01:40","modified_gmt":"2023-07-10T14:01:40","slug":"%e5%b9%b3%e9%9d%a2%e7%9a%84%e9%9a%90%e5%bc%8f%e4%b8%80%e8%88%ac%e6%96%b9%e7%a8%8b%e6%88%96%e7%ac%9b%e5%8d%a1%e5%b0%94%e6%96%b9%e7%a8%8b","status":"publish","type":"post","link":"https:\/\/mathority.org\/cn\/%e5%b9%b3%e9%9d%a2%e7%9a%84%e9%9a%90%e5%bc%8f%e4%b8%80%e8%88%ac%e6%96%b9%e7%a8%8b%e6%88%96%e7%ac%9b%e5%8d%a1%e5%b0%94%e6%96%b9%e7%a8%8b\/","title":{"rendered":"\u5e73\u9762\u7684\u9690\u5f0f\u3001\u4e00\u822c\u6216\u7b1b\u5361\u5c14\u65b9\u7a0b"},"content":{"rendered":"<p>\u89e3\u91ca\u5982\u4f55\u8ba1\u7b97\u9690\u5f0f\u5e73\u9762\u65b9\u7a0b\uff08\u516c\u5f0f\uff09\uff0c\u4e5f\u79f0\u4e3a\u4e00\u822c\u65b9\u7a0b\u6216\u7b1b\u5361\u5c14\u65b9\u7a0b\u3002\u6b64\u5916\uff0c\u60a8\u8fd8\u5c06\u4e86\u89e3\u5982\u4f55\u4ece\u5e73\u9762\u7684\u6cd5\u5411\u91cf\u627e\u5230\u5e73\u9762\u7684\u65b9\u7a0b\u3002\u800c\u4e14\uff0c\u66f4\u91cd\u8981\u7684\u662f\uff0c\u60a8\u5c06\u80fd\u591f\u770b\u5230\u9010\u6b65\u89e3\u51b3\u7684\u793a\u4f8b\u548c\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=\"%c2%bfque-es-la-ecuacion-implicita-o-general-del-plano\"><\/span>\u8be5\u8ba1\u5212\u7684\u9690\u5f0f\u6216\u4e00\u822c\u65b9\u7a0b\u662f\u4ec0\u4e48\uff1f <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-105\"><\/div>\n<\/div>\n<p>\u5728\u89e3\u6790\u51e0\u4f55\u4e2d\uff0c<strong>\u5e73\u9762\u7684\u9690\u5f0f\u65b9\u7a0b<\/strong>\uff0c\u4e5f\u79f0\u4e3a\u5e73\u9762\u7684<strong>\u4e00\u822c<\/strong>\u65b9\u7a0b\u6216<strong>\u7b1b\u5361\u5c14<\/strong>\u65b9\u7a0b\uff0c\u662f\u5141\u8bb8\u4efb\u4f55\u5e73\u9762\u4ee5\u6570\u5b66\u65b9\u5f0f\u8868\u8fbe\u7684\u65b9\u7a0b\u3002\u4e3a\u4e86\u627e\u5230\u5e73\u9762\u7684\u9690\u5f0f\u6216\u4e00\u822c\u65b9\u7a0b\uff0c\u6211\u4eec\u9700\u8981\u4e00\u4e2a\u70b9\u548c\u4e24\u4e2a\u5c5e\u4e8e\u8be5\u5e73\u9762\u7684\u7ebf\u6027\u72ec\u7acb\u5411\u91cf\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-ecuacion-implicita-o-general-del-plano\"><\/span>\u8ba1\u5212\u7684\u9690\u5f0f\u6216\u4e00\u822c\u65b9\u7a0b\u7684\u516c\u5f0f<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<div style=\"background-color:#FFCC8080;padding-top: 20px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px solid #FFB74D; border-radius:20px;\">\n<p style=\"text-align:left\">\u8003\u8651\u5e73\u9762\u7684\u4e00\u4e2a\u70b9\u548c\u4e24\u4e2a\u65b9\u5411\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-cf5d4130501bb01b15aa80f8f80caf1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c} P(P_x,P_y,P_z) \\\\[2ex] \\vv{\\text{u}}=(\\text{u}_x,\\text{u}_y,\\text{u}_z)\\\\[2ex] \\vv{\\text{v}}=(\\text{v}_x,\\text{v}_y,\\text{v}_z)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"95\" width=\"116\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\">\u5e73\u9762\u7684\u9690\u5f0f\u3001\u4e00\u822c\u6216\u7b1b\u5361\u5c14\u65b9\u7a0b\u901a\u8fc7\u6c42\u89e3\u4ee5\u4e0b\u884c\u5217\u5f0f\u5e76\u5c06\u7ed3\u679c\u8bbe\u7f6e\u4e3a 0 \u6765\u83b7\u5f97\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-68d67612dfa54d76666aa37b702a472f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix}\\text{u}_x &amp; \\text{v}_x &amp; x-P_x \\\\[1.1ex]\\text{u}_y &amp; \\text{v}_y &amp; y-P_y \\\\[1.1ex]\\text{u}_z &amp; \\text{v}_z &amp; z-P_z \\end{vmatrix} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"163\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p style=\"text-align:left;\">\u56e0\u6b64\uff0c<strong>\u6240\u5f97\u8ba1\u5212\u7684\u9690\u5f0f\u6216\u4e00\u822c\u65b9\u7a0b<\/strong>\u5982\u4e0b\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-27e298e3103f917bd81b20315b6d9025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Ax+By+Cz+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"183\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<\/div>\n<p>\u91cd\u8981\u7684\u662f\uff0c\u516c\u5f0f\u4e2d\u7684\u4e24\u4e2a\u5411\u91cf\u5f7c\u6b64\u7ebf\u6027\u65e0\u5173\uff0c\u5373\u5b83\u4eec\u5fc5\u987b\u5177\u6709\u4e0d\u540c\u7684\u65b9\u5411\u3002\u4e3a\u4e86\u6ee1\u8db3\u8fd9\u4e2a\u6761\u4ef6\uff0c\u4e24\u4e2a\u5411\u91cf\u4e0d\u5e73\u884c\u5c31\u8db3\u591f\u4e86\u3002 <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/equations-planes.webp\" alt=\"r3 \u4e2d\u6cdb xy \u7684\u9690\u5f0f\u6216\u4e00\u822c\u6216\u7b1b\u5361\u5c14\u65b9\u7a0b\" class=\"wp-image-2443\" width=\"393\" height=\"138\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p>\u867d\u7136\u6ca1\u6709\u5fc5\u8981\u77e5\u9053\u8fd9\u4e2a\u516c\u5f0f\u7684\u539f\u56e0\uff0c\u4f46\u60a8\u53ef\u4ee5\u5728\u4e0b\u9762\u770b\u5230\u5b83\u7684\u6f14\u793a\u3002<\/p>\n<p>\u4ece\u8ba1\u5212\u7684\u53c2\u6570\u65b9\u7a0b\u5f00\u59cb\uff0c\u6211\u4eec\u5c06\u7ee7\u7eed\u8ba8\u8bba\u8ba1\u5212\u7684\u9690\u5f0f\uff08\u6216\u4e00\u822c\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-46f87775f11f01a59c70aa3ee864aebe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{cases}x=P_x + \\lambda \\text{u}_x + \\mu \\text{v}_x \\\\[1.7ex] y=P_y + \\lambda \\text{u}_y + \\mu \\text{v}_y\\\\[1.7ex] z=P_z + \\lambda\\text{u}_z + \\mu \\text{v}_z \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"167\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u9996\u5148\uff0c\u6211\u4eec\u5c06\u6bcf\u4e2a\u53c2\u6570\u65b9\u7a0b\u7684\u72ec\u7acb\u9879\u4f20\u9012\u5230\u65b9\u7a0b\u7684\u53e6\u4e00\u8fb9\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-0c2f3831ca03939d7e23d24c7d435337_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{cases}x-P_x= \\lambda \\text{u}_x + \\mu \\text{v}_x \\\\[1.7ex] y-P_y = \\lambda \\text{u}_y + \\mu \\text{v}_y\\\\[1.7ex] z-P_z = \\lambda\\text{u}_z + \\mu \\text{v}_z \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"167\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u6216\u540c\u7b49\u5b66\u5386\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-aca0b16ff9b92401181c2bdc5ba981bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{cases} \\lambda \\text{u}_x + \\mu \\text{v}_x =x-P_x\\\\[1.7ex]  \\lambda \\text{u}_y + \\mu \\text{v}_y=y-P_y \\\\[1.7ex]  \\lambda\\text{u}_z + \\mu \\text{v}_z =z-P_z\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"166\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u4e3a\u4e86\u4f7f\u4e0a\u8ff0\u65b9\u7a0b\u7ec4\u6709\u53ef\u884c\u89e3\uff0c\u4ee5\u4e0b\u77e9\u9635\u7684\u79e9\u5fc5\u987b\u7b49\u4e8e 2\uff08Rouche-Frobenius \u5b9a\u7406\uff09\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-6f802b760ba5ab681afd0f02c83eddb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{pmatrix}\\text{u}_x &amp; \\text{v}_x &amp; x-P_x \\\\[1.1ex]\\text{u}_y &amp; \\text{v}_y &amp; y-P_y \\\\[1.1ex]\\text{u}_z &amp; \\text{v}_z &amp; z-P_z\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"142\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\uff0c\u5982\u679c\u524d\u4e00\u4e2a\u77e9\u9635\u7684\u8303\u56f4\u5fc5\u987b\u4e3a 2\uff0c\u5219 3&#215;3 \u884c\u5217\u5f0f\u5fc5\u987b\u7b49\u4e8e 0\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-917f1770ff2a17897e5df76998ec3519_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{vmatrix}\\text{u}_x &amp; \\text{v}_x &amp; x-P_x \\\\[1.1ex]\\text{u}_y &amp; \\text{v}_y &amp; y-P_y \\\\[1.1ex]\\text{u}_z &amp; \\text{v}_z &amp; z-P_z \\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"163\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u901a\u8fc7\u6c42\u89e3\u8fd9\u4e2a\u884c\u5217\u5f0f\uff0c\u6211\u4eec\u5f97\u5230\u5e73\u9762\u7684\u4e00\u822c\u3001\u9690\u5f0f\u6216\u7b1b\u5361\u5c14\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-27e298e3103f917bd81b20315b6d9025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Ax+By+Cz+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"183\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p>\u6240\u4ee5\uff0c\u6211\u4eec\u521a\u521a\u770b\u5230\u4e86\u5e73\u9762\u7684\u9690\u5f0f\uff08\u6216\u4e00\u822c\uff09\u65b9\u7a0b\u548c\u53c2\u6570\u65b9\u7a0b\uff0c\u4f46\u662f\uff0c\u8fd8\u6709\u66f4\u591a\u7684\u65b9\u6cd5\u53ef\u4ee5\u89e3\u6790\u5730\u8868\u8fbe\u5e73\u9762\uff0c\u4f8b\u5982\u77e2\u91cf\u65b9\u7a0b\u548c\u89c4\u8303\u65b9\u7a0b\u3002\u60a8\u53ef\u4ee5\u5728\u6b64\u94fe\u63a5\u4e2d\u67e5\u770b<a href=\"https:\/\/mathority.org\/cn\/\u7a7a\u95f4\u5e73\u9762\u65b9\u7a0b\/\">\u8ba1\u5212\u4e2d\u6240\u6709\u65b9\u7a0b<\/a>\u7684\u516c\u5f0f\u548c\u89e3\u91ca\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-como-hallar-la-ecuacion-implicita-o-general-del-plano\"><\/span>\u5982\u4f55\u627e\u5230\u5e73\u9762\u7684\u9690\u5f0f\u6216\u4e00\u822c\u65b9\u7a0b\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-109\"><\/div>\n<\/div>\n<p>\u6211\u4eec\u901a\u8fc7\u4e00\u4e2a\u4f8b\u5b50\u6765\u770b\u770b\u5982\u4f55\u786e\u5b9a\u5e73\u9762\u7684\u9690\u5f0f\uff08\u6216\u4e00\u822c\u6216\u7b1b\u5361\u5c14\uff09\u65b9\u7a0b\uff1a<\/p>\n<ul>\n<li>\u6c42\u901a\u8fc7\u8be5\u70b9\u7684\u5e73\u9762\u7684\u9690\u5f0f\u6216\u4e00\u822c\u65b9\u7a0b\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2118d788ea1b10a70b36f284857de70e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(3,1,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u5e76\u5305\u542b\u5411\u91cf<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef85b7409667d52ad3c9f8981dad5f7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}=(2,0,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"89\" 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-a50e73125c9dcf4a9c58d316fbf850ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}=(4,-1,2).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<p>\u4e3a\u4e86\u8ba1\u7b97\u5e73\u9762\u7684\u4e00\u822c\u6216\u9690\u5f0f\u65b9\u7a0b\uff0c\u9700\u8981\u6c42\u89e3\u7531\u4e24\u4e2a\u5411\u91cf\u3001\u53d8\u91cf\u548c\u70b9\u7684\u5750\u6807\u5f62\u6210\u7684\u4ee5\u4e0b\u884c\u5217\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-917f1770ff2a17897e5df76998ec3519_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{vmatrix}\\text{u}_x &amp; \\text{v}_x &amp; x-P_x \\\\[1.1ex]\\text{u}_y &amp; \\text{v}_y &amp; y-P_y \\\\[1.1ex]\\text{u}_z &amp; \\text{v}_z &amp; z-P_z \\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"163\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\uff0c\u6211\u4eec\u5c06\u5411\u91cf\u548c\u70b9\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-aa25223c3a00e31f89043a3500d32c68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{vmatrix}2 &amp; 4 &amp; x-3 \\\\[1.1ex]0 &amp; -1 &amp; y-1 \\\\[1.1ex]3&amp; 2 &amp; z-(-1) \\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"173\" 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-7886f1c2758c204802b96f44acc8a7cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{vmatrix}2 &amp; 4 &amp; x-3 \\\\[1.1ex]0 &amp; -1 &amp; y-1 \\\\[1.1ex]3&amp; 2 &amp; z+1 \\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"147\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u73b0\u5728\u6211\u4eec\u6c42\u89e3 3 \u9636\u884c\u5217\u5f0f\uff0c\u4f8b\u5982\u4f7f\u7528 Sarrus \u89c4\u5219\u6216\u901a\u8fc7\u8f85\u56e0\u5b50\uff08\u6216\u526f\u56e0\u5b50\uff09\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-3d8c2ae50c69efa75c6f439ec502a6d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-2(z+1)+12(y-1)+3(x-3)-4(y-1) = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"370\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u73b0\u5728\u6211\u4eec\u5bf9\u672f\u8bed\u8fdb\u884c\u64cd\u4f5c\u548c\u5206\u7ec4\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-e9cfe2d167a92bc1e64737ce9e7a5ed2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3(x-3)+8(y-1) -2(z+1) = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"264\" 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-3cf3da637bf3a8d04fb2b8d791d78a9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x-9+8y-8 -2z-2 = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"223\" 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-65167b5921553dd9e11f9ac9bb80864e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x+8y-2z-19 = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"170\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\uff0c\u8be5\u8ba1\u5212\u7684\u9690\u5f0f\u6216\u4e00\u822c\u65b9\u7a0b\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-152a3c9fd09fb72b6269201f02fb8303_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{3x+8y-2z-19 = 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"170\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"calcular-la-ecuacion-implicita-o-general-de-un-plano-a-partir-de-su-vector-normal\"><\/span>\u4ece\u5e73\u9762\u7684\u6cd5\u5411\u91cf\u8ba1\u7b97\u5e73\u9762\u7684\u9690\u5f0f\u6216\u4e00\u822c\u65b9\u7a0b<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u5e73\u9762\u65b9\u7a0b\u4e2d\u4e00\u4e2a\u975e\u5e38\u5178\u578b\u7684\u95ee\u9898\u662f\u5728\u7ed9\u5b9a\u70b9\u53ca\u5176\u6cd5\u5411\uff08\u6216\u5782\u76f4\uff09\u5411\u91cf\u7684\u60c5\u51b5\u4e0b\u627e\u5230\u7ed9\u5b9a\u5e73\u9762\u7684\u65b9\u7a0b\u3002\u90a3\u4e48\uff0c\u8ba9\u6211\u4eec\u770b\u770b\u5b83\u662f\u5982\u4f55\u5de5\u4f5c\u7684\u3002<\/p>\n<p>\u4f46\u60a8\u5fc5\u987b\u9996\u5148\u77e5\u9053<strong>\u5782\u76f4\u4e8e\u5e73\u9762\u7684\u5411\u91cf\u7684\u5206\u91cf X\u3001Y\u3001Z<strong>\u5206\u522b<\/strong><\/strong><strong>\u4e0e\u8be5\u5e73\u9762\u7684\u9690\u5f0f\uff08\u6216\u4e00\u822c\uff09\u65b9\u7a0b\u7684\u7cfb\u6570 A\u3001B\u3001C \u4e00\u81f4\u3002<\/strong><\/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-27f3ee5d7e81864550f3b86fdd53e89d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\color{orange} \\boxed{ \\color{black} \\quad \\pi : \\ Ax+By+C+D = 0 \\quad \\iff \\quad \\vv{n} = (A,B,C) \\quad \\vphantom{\\Bigl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"540\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p>\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-10affe1faee06a5faa4ef6d9c0473b1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p>\u662f\u4e0e\u5e73\u9762\u6b63\u4ea4\u7684\u5411\u91cf<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-26622dd58bf71cd1b543c3d83233c561_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u4e00\u65e6\u6211\u4eec\u77e5\u9053\u4e86\u524d\u9762\u7684\u5173\u7cfb\uff0c\u8ba9\u6211\u4eec\u770b\u4e00\u4e2a\u89e3\u51b3\u6b64\u7c7b\u5e73\u9762\u65b9\u7a0b\u95ee\u9898\u7684\u4f8b\u5b50\uff1a<\/p>\n<ul>\n<li>\u786e\u5b9a\u901a\u8fc7\u8be5\u70b9\u7684\u5e73\u9762\u7684\u9690\u5f0f\u6216\u4e00\u822c\u65b9\u7a0b\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6df0e548515bb2b24f352853a2614015_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1,0,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u5b83\u7684\u6cd5\u5411\u91cf\u4e4b\u4e00\u662f<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-44298f830c420011a4326017f5fd7cfb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}=(3,-1,2) .\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<p>\u5e73\u9762\u7684\u9690\u5f0f\u3001\u4e00\u822c\u6216\u7b1b\u5361\u5c14\u65b9\u7a0b\u7684\u516c\u5f0f\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-27e298e3103f917bd81b20315b6d9025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Ax+By+Cz+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"183\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\uff0c\u4ece\u6cd5\u5411\u91cf\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u627e\u5230\u7cfb\u6570 A\u3001B \u548c C\uff0c\u56e0\u4e3a\u5b83\u4eec\u76f8\u5f53\u4e8e\u6cd5\u5411\u91cf\u7684\u5206\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-596b3fdc65160234c06b0d28aebea74f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}=(3,-1,2) \\ \\longrightarrow \\ 3x-1y+2z+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"322\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u800c\u6211\u4eec\u53ea\u9700\u8981\u627e\u5230\u53c2\u6570D\u5373\u53ef\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u5c06\u5c5e\u4e8e\u5e73\u9762\u7684\u70b9\u7684\u5750\u6807\u4ee3\u5165\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-6df0e548515bb2b24f352853a2614015_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1,0,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" 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-09a0b099394ba4634fb6d7aad3a627e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3\\cdot 1-0+2\\cdot (-2)+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"210\" 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-b90d57a1708925626a300c7e5db673ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3-4+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"109\" 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-d2dfb99a77bfcee1b72540db9cf579a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-1+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"91\" 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-16ace3f6683252e5630a1091bbc0404e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D=1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\u8be5\u8ba1\u5212\u7684\u9690\u5f0f\u6216\u4e00\u822c\u65b9\u7a0b\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-cde6db2f935eb7d0ad39141f86aab013_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{3x-y+2z+1 = 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"153\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-ecuacion-implicita-o-general-del-plano\"><\/span>\u89e3\u51b3\u4e86\u5e73\u9762\u9690\u5f0f\u6216\u4e00\u822c\u65b9\u7a0b\u7684\u95ee\u9898<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e601<\/h3>\n<p>\u6c42\u901a\u8fc7\u8be5\u70b9\u7684\u5e73\u9762\u7684\u9690\u5f0f\u6216\u4e00\u822c\u65b9\u7a0b<\/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-b61f338575699a918d594595ddc6fb02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-2,1,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u5e76\u5305\u542b\u5411\u91cf<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5150cc543174c7c079a38b016685eb3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}=(4,1,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"89\" 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-0664398c6f1938e4ec5efaae48ab1c70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}=(5,3,-1).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" 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\">\u4e3a\u4e86\u8ba1\u7b97\u5e73\u9762\u7684\u4e00\u822c\u6216\u9690\u5f0f\u65b9\u7a0b\uff0c\u9700\u8981\u6c42\u89e3\u7531\u4e24\u4e2a\u5411\u91cf\u3001\u4e09\u4e2a\u53d8\u91cf\u548c\u70b9\u7684\u5750\u6807\u7ec4\u6210\u7684\u4ee5\u4e0b\u884c\u5217\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-917f1770ff2a17897e5df76998ec3519_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{vmatrix}\\text{u}_x &amp; \\text{v}_x &amp; x-P_x \\\\[1.1ex]\\text{u}_y &amp; \\text{v}_y &amp; y-P_y \\\\[1.1ex]\\text{u}_z &amp; \\text{v}_z &amp; z-P_z \\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"163\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\uff0c\u6211\u4eec\u5c06\u5411\u91cf\u548c\u70b9\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-02e103601cd9992a8a8c087d016a08c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{vmatrix}4 &amp; 5 &amp; x+2 \\\\[1.1ex]1 &amp; 3 &amp; y-1 \\\\[1.1ex]3&amp; 1 &amp; z+1 \\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"133\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u73b0\u5728\u6211\u4eec\u7528\u60a8\u9009\u62e9\u7684\u65b9\u6cd5\u6c42\u89e3 3\u00d73 \u77e9\u9635\u7684\u884c\u5217\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-ec35e71b9cca25aa9907c97da2ea2e2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"12(z+1)+15(y-1)+1(x+2)-9(x+2)-4(y-1)-5(z+1) = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"534\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u6211\u4eec\u6267\u884c\u64cd\u4f5c\u5e76\u5bf9\u76f8\u4f3c\u9879\u8fdb\u884c\u5206\u7ec4\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-fea5513ce57c88cac89a695b47b7a0c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-8(x+2)+11(y-1)+7(z+1) = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"286\" 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-b965060b2baad7f1b7fa66e0555a23f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-8x-16+11y-11+7z+7=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"262\" 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-52635b24c2d396ff9a47fbd210c56bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-8x+11y+7z-20= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"192\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\u8be5\u8ba1\u5212\u7684\u9690\u5f0f\u6216\u4e00\u822c\u65b9\u7a0b\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-fc97887dad7e25ff823eee155fb58358_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{-8x+11y+7z-20 = 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"192\" style=\"vertical-align: -4px;\"><\/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>\u5224\u65ad\u70b9\u662f\u5426<\/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-10de27107c1cf63dc889433e271d4a78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-1,5,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u5c5e\u4e8e\u4ee5\u4e0b\u8ba1\u5212\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-54a545de55c14f27d77bfda0188789a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi : \\ 2x+y+6z-5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"184\" 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\">\u5bf9\u4e8e\u4f4d\u4e8e\u5e73\u9762\u4e0a\u7684\u70b9\uff0c\u5fc5\u987b\u9a8c\u8bc1\u5176\u65b9\u7a0b\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u5c06\u70b9\u7684\u7b1b\u5361\u5c14\u5750\u6807\u4ee3\u5165\u5e73\u9762\u7684\u65b9\u7a0b\u4e2d\uff0c\u5e76\u68c0\u67e5\u65b9\u7a0b\u662f\u5426\u6210\u7acb\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-25ef56f6218537e6592c6ce17e0c3cb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x+y+6z-5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"153\" 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-10de27107c1cf63dc889433e271d4a78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-1,5,-3)\" 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-b30e787cb2dfac5e87b0758b866e10a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2\\cdot (-1)+5+6\\cdot (-3)-5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"232\" 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-66acebc4d56a2fb70fe23b2ead99dcc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-2+5-18-5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"155\" 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-64706704bc6f3ef930293722159a8861_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-20\\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"63\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u8be5\u70b9\u4e0d\u9075\u5b88\u5e73\u9762\u65b9\u7a0b\uff0c<strong>\u56e0\u6b64\u5b83\u4e0d\u662f\u8be5\u5e73\u9762\u7684\u4e00\u90e8\u5206\u3002<\/strong><\/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>\u627e\u5230\u5305\u542b\u4ee5\u4e0b\u4e09\u70b9\u7684\u8ba1\u5212\u7684\u9690\u5f0f\uff08\u6216\u4e00\u822c\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-c3003c4a812ae18bebda7f61c0bbe5f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(5,-1,-2) \\qquad B(2,1,3) \\qquad C(4,1,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"322\" 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\">\u4e3a\u4e86\u627e\u5230\u5e73\u9762\u7684\u9690\u5f0f\u65b9\u7a0b\uff0c\u6211\u4eec\u9700\u8981\u627e\u5230\u4e24\u4e2a\u7ed1\u5b9a\u5728\u5e73\u9762\u4e0a\u7684\u7ebf\u6027\u65e0\u5173\u5411\u91cf\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u8ba1\u7b97\u7531 3 \u4e2a\u70b9\u5b9a\u4e49\u7684\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-f96add9ffd85a60c66b7b40a65192537_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB} = B - A = (2,1,3) - (5,-1,-2) = (-3,2,5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"379\" 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-09c96a38b66a9b0108da71a2c7ea0b2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AC} = C - A = (4,1,-2) - (5,-1,-2) = (-1,2,0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"392\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u627e\u5230\u7684\u4e24\u4e2a\u5411\u91cf\u7684\u5750\u6807\u4e0d\u6210\u6bd4\u4f8b\uff0c\u56e0\u6b64\u5b83\u4eec\u5b9e\u9645\u4e0a\u662f\u5f7c\u6b64\u7ebf\u6027\u72ec\u7acb\u7684\u3002<\/p>\n<p class=\"has-text-align-left\">\u73b0\u5728\u6211\u4eec\u5df2\u7ecf\u77e5\u9053\u4e86\u5e73\u9762\u4e0a\u7684\u4e24\u4e2a\u65b9\u5411\u5411\u91cf\u548c\u4e00\u4e2a\u70b9\uff0c\u56e0\u6b64\u6211\u4eec\u5df2\u7ecf\u53ef\u4ee5\u5e94\u7528\u5e73\u9762\u4e00\u822c\u65b9\u7a0b\u7684\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-917f1770ff2a17897e5df76998ec3519_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{vmatrix}\\text{u}_x &amp; \\text{v}_x &amp; x-P_x \\\\[1.1ex]\\text{u}_y &amp; \\text{v}_y &amp; y-P_y \\\\[1.1ex]\\text{u}_z &amp; \\text{v}_z &amp; z-P_z \\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"163\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6211\u4eec\u5c06\u5411\u91cf\u548c\u4e09\u4e2a\u70b9\u4e4b\u4e00\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-383093e607bc8ecc5f99e1815242b22a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{vmatrix}-3 &amp; -1 &amp; x-5 \\\\[1.1ex]2 &amp; 2 &amp; y+1 \\\\[1.1ex]5&amp; 0 &amp; z+2 \\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"161\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u6211\u4eec\u6c42\u89e3\u884c\u5217\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-a3d7401055c280280af3a3d0486cdb86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-6(z+2)-5(y+1)-10(x-5)+2(z+2)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"370\" 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-4e355f06e9bd200804e0a37e059a389c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-10(x-5)-5(y+1)-4(z+2)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"286\" 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-0f005d4a2fdd910e746e7231a1c83f61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-10x+50-5y-5-4z-8=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"253\" 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-7653b13f379b9d38230f457c59d1a568_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-10x-5y-4z+37=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"192\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u7b80\u800c\u8a00\u4e4b\uff0c\u6240\u8ba8\u8bba\u7684\u5e73\u9762\u7684\u9690\u5f0f\u3001\u4e00\u822c\u6216\u7b1b\u5361\u5c14\u65b9\u7a0b\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-ac148d30ada4a3e256874712b4b196db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{-10x-5y-4z+37=0}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"192\" style=\"vertical-align: -4px;\"><\/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<div class=\"wp-block-group\">\n<div class=\"wp-block-group__inner-container is-layout-flow\">\n<p>\u8ba1\u7b97\u901a\u8fc7\u8be5\u70b9\u7684\u7a7a\u95f4\u5e73\u9762\u7684\u9690\u5f0f\u6216\u4e00\u822c\u65b9\u7a0b<\/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-e42bc8fa114f50a19858a526eabb6e30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(3,4,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u5b83\u7684\u6cd5\u5411\u91cf\u4e4b\u4e00\u662f<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b007924d3ec5c7cd5de5d1d46cc86711_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}=(5,-2,-3) .\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" 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\">\u5e73\u9762\u7684\u9690\u5f0f\u3001\u4e00\u822c\u6216\u7b1b\u5361\u5c14\u65b9\u7a0b\u7684\u516c\u5f0f\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-27e298e3103f917bd81b20315b6d9025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Ax+By+Cz+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"183\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u90a3\u4e48\uff0c\u4ece\u6cd5\u5411\u91cf\u4e2d\u6211\u4eec\u53ef\u4ee5\u627e\u5230\u7cfb\u6570 A\u3001B \u548c C\uff0c\u56e0\u4e3a\u5b83\u4eec\u5206\u522b\u7b49\u4e8e\u6cd5\u5411\u91cf\u7684\u5206\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-8f4b71cc8d8c1610a5d5706ac44d1ad3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}=(5,-2,-3) \\ \\longrightarrow \\ 5x-2y-3z+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"336\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6240\u4ee5\u6211\u4eec\u53ea\u9700\u8981\u627e\u5230\u53c2\u6570D\u5373\u53ef\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u5c06\u5c5e\u4e8e\u5e73\u9762\u7684\u70b9\u7684\u5750\u6807\u4ee3\u5165\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-e42bc8fa114f50a19858a526eabb6e30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(3,4,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" 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-720c689bc073e6c0f865bf406d92cbba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"5\\cdot 3-2\\cdot 4-3\\cdot (-3)+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"232\" 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-3edf41b26a3b9c8bdead74d05767ec60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"15-8+9+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"147\" 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-0db457697a3637eb92bff90460d8e98f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"16+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"86\" 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-a2efb3d90e15e4474268d6de0570da4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D=-16\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"71\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u603b\u4e4b\uff0c\u8be5\u8ba1\u5212\u7684\u9690\u5f0f\u6216\u4e00\u822c\u65b9\u7a0b\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-ee0c114a24764dd24df9d58aab7155ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{5x-2y-3z-16 = 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"170\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u89e3\u91ca\u5982\u4f55\u8ba1\u7b97\u9690\u5f0f\u5e73\u9762\u65b9\u7a0b\uff08\u516c\u5f0f\uff09\uff0c\u4e5f\u79f0\u4e3a\u4e00\u822c\u65b9\u7a0b\u6216\u7b1b\u5361\u5c14\u65b9\u7a0b\u3002\u6b64\u5916\uff0c\u60a8\u8fd8\u5c06\u4e86\u89e3\u5982\u4f55\u4ece\u5e73\u9762\u7684\u6cd5\u5411\u91cf\u627e\u5230\u5e73\u9762\u7684\u65b9\u7a0b &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" 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