{"id":272,"date":"2023-07-10T00:23:44","date_gmt":"2023-07-10T00:23:44","guid":{"rendered":"https:\/\/mathority.org\/cn\/%e7%a9%ba%e9%97%b4%e4%b8%ad%e7%9b%b4%e7%ba%bf%e5%88%b0%e5%b9%b3%e9%9d%a2%e7%9a%84%e8%b7%9d%e7%a6%bb%e5%85%ac%e5%bc%8f\/"},"modified":"2023-07-10T00:23:44","modified_gmt":"2023-07-10T00:23:44","slug":"%e7%a9%ba%e9%97%b4%e4%b8%ad%e7%9b%b4%e7%ba%bf%e5%88%b0%e5%b9%b3%e9%9d%a2%e7%9a%84%e8%b7%9d%e7%a6%bb%e5%85%ac%e5%bc%8f","status":"publish","type":"post","link":"https:\/\/mathority.org\/cn\/%e7%a9%ba%e9%97%b4%e4%b8%ad%e7%9b%b4%e7%ba%bf%e5%88%b0%e5%b9%b3%e9%9d%a2%e7%9a%84%e8%b7%9d%e7%a6%bb%e5%85%ac%e5%bc%8f\/","title":{"rendered":"\u7a7a\u95f4\u4e2d\u76f4\u7ebf\u5230\u5e73\u9762\u7684\u8ddd\u79bb"},"content":{"rendered":"<p>\u5728\u8fd9\u91cc\uff0c\u60a8\u5c06\u4e86\u89e3\u5982\u4f55\u8ba1\u7b97\u4ece\u76f4\u7ebf\u5230\u5e73\u9762\u7684\u8ddd\u79bb\uff0c\u6b64\u5916\uff0c\u60a8\u8fd8\u53ef\u4ee5\u770b\u5230\u9010\u6b65\u89e3\u51b3\u7684\u793a\u4f8b\u548c\u7ec3\u4e60\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcual-es-la-distancia-de-una-recta-a-un-plano\"><\/span>\u76f4\u7ebf\u548c\u5e73\u9762\u4e4b\u95f4\u7684\u8ddd\u79bb\u662f\u591a\u5c11\uff1f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u5728\u89e3\u6790\u51e0\u4f55\u4e2d\uff0c\u7a7a\u95f4\u4e2d\u76f4\u7ebf\u548c\u5e73\u9762\u4e4b\u95f4\u7684\u8ddd\u79bb\u53d6\u51b3\u4e8e\u8fd9\u4e24\u4e2a\u51e0\u4f55\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e\uff1a<\/p>\n<ul>\n<li>\u5982\u679c<strong>\u76f4\u7ebf\u5305\u542b\u5728\u5e73\u9762\u5185<\/strong>\u6216\u8005<strong>\u76f4\u7ebf\u548c\u5e73\u9762\u5e73\u884c<\/strong>\uff0c\u5219\u5b83\u4eec\u4e4b\u95f4\u7684\u8ddd\u79bb\u4e3a\u96f6\u3002<\/li>\n<li>\u5982\u679c<strong>\u76f4\u7ebf\u4e0e\u5e73\u9762\u5e73\u884c<\/strong>\uff0c\u5219\u901a\u8fc7\u53d6\u76f4\u7ebf\u4e0a\u7684\u4efb\u610f\u4e00\u70b9\u5e76\u8ba1\u7b97\u8be5\u70b9\u5230\u5e73\u9762\u7684\u8ddd\u79bb\u5373\u53ef\u627e\u5230\u76f4\u7ebf\u5230\u5e73\u9762\u7684\u8ddd\u79bb\u3002 <\/li>\n<\/ul>\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\/distance-entre-une-ligne-et-un-plan.webp\" alt=\"\u7a7a\u95f4\u4e2d\u4e00\u6761\u7ebf\u5230\u4e00\u4e2a\u5e73\u9762\u7684\u8ddd\u79bb\" class=\"wp-image-4062\" width=\"459\" height=\"252\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p>\u56e0\u6b64\uff0c\u8981\u8ba1\u7b97\u76f4\u7ebf\u5230\u5e73\u9762\u7684\u8ddd\u79bb\uff0c\u5fc5\u987b\u77e5\u9053\u5982\u4f55\u786e\u5b9a<a href=\"https:\/\/mathority.org\/cn\/\u76f4\u7ebf\u548c\u5e73\u9762\u7684\u76f8\u5bf9\u4f4d\u7f6e\/\">\u76f4\u7ebf\u548c\u5e73\u9762\u4e4b\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e<\/a>\u4ee5\u53ca\u5982\u4f55\u8ba1\u7b97<a href=\"https:\/\/mathority.org\/cn\">\u70b9\u548c\u5e73\u9762\u4e4b\u95f4\u7684\u8ddd\u79bb<\/a>\u3002\u56e0\u6b64\uff0c\u5982\u679c\u60a8\u4e0d\u5b8c\u5168\u6e05\u695a\u6216\u4e0d\u77e5\u9053\u516c\u5f0f\uff0c\u6211\u4eec\u5efa\u8bae\u60a8\u9996\u5148\u67e5\u770b\u94fe\u63a5\u7684\u9875\u9762\uff0c\u60a8\u5c06\u5728\u5176\u4e2d\u627e\u5230\u89e3\u91ca\u3001\u793a\u4f8b\u548c\u9010\u6b65\u89e3\u51b3\u7684\u7ec3\u4e60\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-como-calcular-la-distancia-entre-una-recta-y-un-plano\"><\/span>\u8ba1\u7b97\u76f4\u7ebf\u548c\u5e73\u9762\u4e4b\u95f4\u7684\u8ddd\u79bb\u7684\u793a\u4f8b<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u4e3a\u4e86\u8ba9\u60a8\u4e86\u89e3\u5982\u4f55\u6c42\u7a7a\u95f4\u4e2d\u76f4\u7ebf\u548c\u5e73\u9762\u4e4b\u95f4\u7684\u8ddd\u79bb\uff08\u5728 R3 \u4e2d\uff09\uff0c\u6211\u4eec\u5c06\u89e3\u51b3\u4e00\u4e2a\u95ee\u9898\u4f5c\u4e3a\u793a\u4f8b\uff1a<\/p>\n<ul>\n<li>\u7ebf\u8def\u6709\u591a\u8fdc\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p>\u5728\u98de\u673a\u4e0a<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-26d6788550ffd50fe94542bb3e8ee615_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p>\uff1f<\/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-4785f2386b4ff9387b56cf8c5645c62d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle r: \\ \\begin{cases}x=-2+t \\\\[1.7ex] y=1-3t \\\\[1.7ex] z=-1+2t\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"137\" 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-5d8872d79c59216f4a8489f75a751cdb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi : \\ 4x+2y+z-6=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"184\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p>\u8981\u6c42\u76f4\u7ebf\u548c\u5e73\u9762\u4e4b\u95f4\u7684\u8ddd\u79bb\uff0c\u9996\u5148\u5fc5\u987b\u77e5\u9053\u4e24\u8005\u4e4b\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e\u3002<\/p>\n<p>\u4e00\u65b9\u9762\uff0c\u76f4\u7ebf\u4ee5\u53c2\u6570\u65b9\u7a0b\u7684\u5f62\u5f0f\u5b9a\u4e49\uff0c\u6240\u4ee5\u5b83\u7684\u65b9\u5411\u5411\u91cf\u548c\u5b83\u7ecf\u8fc7\u7684\u70b9\u662f\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-0765b71f9eead3b644fd5de150cdf321_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle r: \\ \\begin{cases} \\vv{\\text{v}}_r =(1,-3,2) \\\\[2ex] P(-2,1,-1) \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"154\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u53e6\u4e00\u65b9\u9762\uff0c\u5782\u76f4\u4e8e\u5e73\u9762\u7684\u77e2\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-f4fe9c16b8d08778c89ab62afcd8e34e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n} =(4,2,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"90\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\uff0c\u8981\u786e\u5b9a\u5e73\u9762\u548c\u76f4\u7ebf\u7684\u76f8\u5bf9\u4f4d\u7f6e\uff0c\u9700\u8981\u8ba1\u7b97\u76f4\u7ebf\u7684\u65b9\u5411\u5411\u91cf\u548c\u5e73\u9762\u6cd5\u7ebf\u5411\u91cf\u4e4b\u95f4\u7684\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-39d2d450c32a6fdf3182a3d16cc18116_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\vv{\\text{v}}_r \\cdot \\vv{n} &amp; = (1,-3,2) \\cdot (4,2,1) \\\\[2ex] &amp; = 1 \\cdot 4-3 \\cdot 2 +2\\cdot 1 \\\\[2ex] &amp;= 4 -6 +2 \\\\[2ex] &amp; = 0\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"140\" width=\"202\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u70b9\u79ef\u7684\u7ed3\u679c\u4e3a\u96f6\uff0c\u56e0\u6b64\u76f4\u7ebf\u53ea\u80fd\u5305\u542b\u5728\u5e73\u9762\u5185\u6216\u4e0e\u5e73\u9762\u5e73\u884c\u3002\u56e0\u6b64\uff0c\u4e3a\u4e86\u627e\u51fa\u662f\u54ea\u79cd\u60c5\u51b5\uff0c\u6211\u4eec\u5c06\u76f4\u7ebf\u4e0a\u7684\u70b9\u7684\u7b1b\u5361\u5c14\u5750\u6807\u4ee3\u5165\u5e73\u9762\u7684\u65b9\u7a0b\u4e2d\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-e4916138781ab0b14c0b827b212fdd98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4x+2y+z-6=0 \\ \\xrightarrow{P(-2,1,-1)} \\ 4\\cdot (-2) +2\\cdot 1 -1-6= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"460\" 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-15f3d9b73d0b2646f6d0173989e7b20a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-8+2-1-6 = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"146\" 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-d93d0450119c6530b10ed45266e017e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-13 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"63\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p>\u5c06\u76f4\u7ebf\u4e0a\u7684\u70b9\u4ee3\u5165\u5e73\u9762\u65b9\u7a0b\u4e2d\uff0c\u6211\u4eec\u5f97\u5230\u4e00\u4e2a\u4e0d\u7b49\u5f0f\uff0c\u56e0\u6b64\u8be5\u70b9\u4e0d\u9075\u5b88\u5e73\u9762\u65b9\u7a0b\uff0c\u56e0\u6b64\u76f4\u7ebf\u548c\u5e73\u9762\u662f\u5e73\u884c\u7684\u3002<\/p>\n<p>\u4e00\u65e6\u6211\u4eec\u77e5\u9053\u76f4\u7ebf\u548c\u5e73\u9762\u5e73\u884c\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u8ba1\u7b97\u5b83\u4eec\u4e4b\u95f4\u7684\u51e0\u4f55\u8ddd\u79bb\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u53d6\u76f4\u7ebf\u4e0a\u7684\u70b9\u5e76\u8ba1\u7b97\u4ece\u8be5\u70b9\u5230\u5e73\u9762\u7684\u8ddd\u79bb\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-4413892ee4946e8c956d69a16a8d43ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-2,1,-1) \\qquad \\qquad \\pi: \\ 4x+2y+z-6=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"353\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u56e0\u6b64\uff0c\u6211\u4eec\u4f7f\u7528\u70b9\u5230\u5e73\u9762\u7684\u8ddd\u79bb\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-7fa5cd77905fa8723448de603b90b705_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,\\pi) = \\cfrac{\\lvert A\\cdot x_0+B\\cdot y_0+C\\cdot z_0+D\\rvert}{\\sqrt{A^2+B^2+C^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"300\" style=\"vertical-align: -16px;\"><\/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-beb8f4629b0195d8ea0870d678585895_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,\\pi) = \\cfrac{\\lvert 4\\cdot (-2)+2\\cdot 1+1\\cdot (-1)-6\\rvert}{\\sqrt{4^2+2^2+1^2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"310\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p>\u6700\u540e\uff0c\u6211\u4eec\u6267\u884c\u4ee5\u4e0b\u64cd\u4f5c\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-570743258f771c18a79b360b80b6a462_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,\\pi) = \\cfrac{\\lvert -8+2-1-6\\rvert}{\\sqrt{16+4+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"204\" style=\"vertical-align: -16px;\"><\/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-3ea97bcf06817734423f2e6e662a1658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,\\pi) = \\cfrac{\\lvert -13\\rvert}{\\sqrt{21}}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"121\" style=\"vertical-align: -16px;\"><\/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-0a19bec182b20b9b1a5a08b41cd817c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"d(P,\\pi) = \\cfrac{13}{\\sqrt{21}}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"112\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p>\u8fd9\u6837\u76f4\u7ebf\u5230\u5e73\u9762\u7684\u8ddd\u79bb\u5c31\u76f8\u5f53\u4e8e\u8ba1\u7b97\u51fa\u7684\u70b9\u5230\u5e73\u9762\u7684\u8ddd\u79bb\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-82d2055df31134bca62040c367778b50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{d(r,\\pi)=d(P,\\pi) = }\\cfrac{\\bm{13}}{\\bm{\\sqrt{21}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"180\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p>\u663e\u7136\uff0c\u8ddd\u79bb\u5fc5\u987b\u603b\u662f\u7ed9\u6211\u4eec\u4e00\u4e2a\u6b63\u503c\uff0c\u56e0\u4e3a\u8ddd\u79bb\u603b\u662f\u6b63\u503c\u3002\u5982\u679c\u6211\u4eec\u5f97\u5230\u5426\u5b9a\u7684\u7ed3\u679c\uff0c\u5219\u610f\u5473\u7740\u6211\u4eec\u5728\u91c7\u53d6\u6b65\u9aa4\u65f6\u72af\u4e86\u9519\u8bef\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5728\u8fd9\u91cc\uff0c\u60a8\u5c06\u4e86\u89e3\u5982\u4f55\u8ba1\u7b97\u4ece\u76f4\u7ebf\u5230\u5e73\u9762\u7684\u8ddd\u79bb\uff0c\u6b64\u5916\uff0c\u60a8\u8fd8\u53ef\u4ee5\u770b\u5230\u9010\u6b65\u89e3\u51b3\u7684\u793a\u4f8b\u548c\u7ec3\u4e60\u3002 \u76f4\u7ebf\u548c\u5e73\u9762\u4e4b\u95f4\u7684\u8ddd\u79bb\u662f\u591a &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/cn\/%e7%a9%ba%e9%97%b4%e4%b8%ad%e7%9b%b4%e7%ba%bf%e5%88%b0%e5%b9%b3%e9%9d%a2%e7%9a%84%e8%b7%9d%e7%a6%bb%e5%85%ac%e5%bc%8f\/\"> <span class=\"screen-reader-text\">\u7a7a\u95f4\u4e2d\u76f4\u7ebf\u5230\u5e73\u9762\u7684\u8ddd\u79bb<\/span> \u67e5\u770b\u5168\u6587 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