{"id":214,"date":"2023-07-11T12:27:22","date_gmt":"2023-07-11T12:27:22","guid":{"rendered":"https:\/\/mathority.org\/cn\/%e4%b8%89%e4%b8%aa%e5%90%91%e9%87%8f%e6%88%96%e4%b8%89%e9%87%8d%e6%a0%87%e9%87%8f%e7%a7%af%e7%9a%84%e6%b7%b7%e5%90%88%e7%a7%af%e7%a4%ba%e4%be%8b\/"},"modified":"2023-07-11T12:27:22","modified_gmt":"2023-07-11T12:27:22","slug":"%e4%b8%89%e4%b8%aa%e5%90%91%e9%87%8f%e6%88%96%e4%b8%89%e9%87%8d%e6%a0%87%e9%87%8f%e7%a7%af%e7%9a%84%e6%b7%b7%e5%90%88%e7%a7%af%e7%a4%ba%e4%be%8b","status":"publish","type":"post","link":"https:\/\/mathority.org\/cn\/%e4%b8%89%e4%b8%aa%e5%90%91%e9%87%8f%e6%88%96%e4%b8%89%e9%87%8d%e6%a0%87%e9%87%8f%e7%a7%af%e7%9a%84%e6%b7%b7%e5%90%88%e7%a7%af%e7%a4%ba%e4%be%8b\/","title":{"rendered":"\u4e09\u4e2a\u5411\u91cf\u7684\u6df7\u5408\u79ef\uff08\u6216\u4e09\u91cd\u70b9\u79ef\uff09"},"content":{"rendered":"<p>\u5728\u6b64\u9875\u9762\u4e0a\uff0c\u6211\u4eec\u89e3\u91ca\u4ec0\u4e48\u662f\u4e09\u4e2a\u5411\u91cf\u7684\u6df7\u5408\u79ef\uff08\u6216\u4e09\u91cd\u70b9\u79ef\uff09\u4ee5\u53ca\u5b83\u7684\u8ba1\u7b97\u65b9\u6cd5\u3002\u60a8\u8fd8\u5c06\u770b\u5230\u6709\u5173\u5411\u91cf\u4e4b\u95f4\u6b64\u7c7b\u8fd0\u7b97\u7684\u793a\u4f8b\u3001\u7ec3\u4e60\u548c\u5df2\u89e3\u51b3\u7684\u95ee\u9898\u3002\u6b64\u5916\uff0c\u60a8\u8fd8\u4f1a\u53d1\u73b0\u6df7\u5408\u4ea7\u54c1\u7684\u7279\u6027\u548c\u5e94\u7528\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-el-producto-mixto-de-tres-vectores\"><\/span>\u4e09\u4e2a\u5411\u91cf\u7684\u6df7\u5408\u79ef\u662f\u591a\u5c11\uff1f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u4e09\u4e2a\u5411\u91cf\u7684<strong>\u6df7\u5408\u79ef<\/strong>\uff0c\u4e5f\u79f0\u4e3a<strong>\u4e09\u91cd\u70b9\u79ef<\/strong>\uff0c\u662f\u4e09\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u8fde\u7eed\u4e58\u6cd5\uff0c\u6d89\u53ca\u4e24\u79cd\u4e0d\u540c\u7c7b\u578b\u7684\u8fd0\u7b97\uff1a <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\/\">\u70b9\u79ef<\/a>\u548c<a href=\"https:\/\/mathority.org\/cn\/\u4e24\u4e2a\u5411\u91cf\u7684\u53c9\u79ef\u4ea4\u53c9\u516c\u5f0f\u793a\u4f8b\u5df2\u89e3\u51b3\u7ec3\u4e60\/\">\u5411\u91cf\u79ef<\/a>\u3002\u56e0\u6b64\uff0c\u4e24\u4e2a\u5411\u91cf\u8fd0\u7b97\u7684\u7ec4\u5408\u7ed9\u51fa\u4e00\u4e2a\u6807\u91cf\uff08\u5b9e\u6570\uff09\u3002<\/p>\n<p>\u5177\u4f53\u5730\uff0c\u6df7\u5408\u4e58\u79ef\u5305\u62ec\u8ba1\u7b97\u4e24\u4e2a\u5411\u91cf\u7684\u5411\u91cf\u4e58\u79ef\uff0c\u5e76\u4e14\u968f\u540e\u5c06\u6240\u83b7\u5f97\u7684\u7ed3\u679c\u4e0e\u7b2c\u4e09\u5411\u91cf\u76f8\u4e58\u3002\u8fd9\u6837\u5199\u770b\u8d77\u6765\u53ef\u80fd\u5f88\u590d\u6742\uff0c\u4f46\u5b9e\u9645\u4e0a\u5e76\u6ca1\u6709\u90a3\u4e48\u590d\u6742\uff0c\u770b\u4e00\u4e0b\u4e09\u70b9\u79ef\u516c\u5f0f\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7904652c5b12243a6dc713936dba9d6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bigl[\\vv{\\text{u}},\\vv{\\text{v}},\\vv{\\text{w}}\\bigr] = \\vv{\\text{u}} \\cdot ( \\vv{\\text{v}}\\times \\vv{\\text{w}})\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"163\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p>\u6b63\u5982\u60a8\u5728\u5176\u516c\u5f0f\u4e2d\u770b\u5230\u7684\uff0c\u4e09\u4e2a\u5411\u91cf\u7684\u6df7\u5408\u4e58\u79ef\u7531\u4e24\u4e2a\u65b9\u62ec\u53f7\u8868\u793a\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcomo-calcular-el-producto-mixto-de-tres-vectores\"><\/span>\u5982\u4f55\u8ba1\u7b97\u4e09\u4e2a\u5411\u91cf\u7684\u6df7\u5408\u79ef\uff1f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u4e09\u70b9\u79ef\u516c\u5f0f\u662f\u6211\u4eec\u5728\u4e0a\u4e00\u8282\u4e2d\u770b\u5230\u7684\u516c\u5f0f\uff0c\u4f46\u662f\uff0c\u5b83\u901a\u5e38\u4e0d\u7528\u4e8e\u786e\u5b9a\u4e09\u4e2a\u5411\u91cf\u7684\u6df7\u5408\u79ef\uff0c\u56e0\u4e3a\u8fd8\u6709\u53e6\u4e00\u79cd\u66f4\u7b80\u5355\u3001\u66f4\u5feb\u7684\u65b9\u6cd5\uff1a <\/p>\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\">\u8bbe\u4efb\u610f 3 \u4e2a\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-3d0daa4c1f9e1aff4f64a39f229bc207_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}= (\\text{u}_x,\\text{u}_y,\\text{u}_z) \\qquad \\vv{\\text{v}}= (\\text{v}_x,\\text{v}_y,\\text{v}_z)\\qquad \\vv{\\text{w}}= (\\text{w}_x,\\text{w}_y,\\text{w}_z)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"431\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\">\u8981\u8ba1\u7b97<strong>\u4e09\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u6df7\u5408\u79ef\uff0c<\/strong>\u53ea\u9700\u6c42\u89e3\u7531\u5411\u91cf\u5206\u91cf\u5f62\u6210\u7684 3\u00d73 \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-a3fed933d4d02bb5ca6f5bae06ea544d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bigl[\\vv{\\text{u}},\\vv{\\text{v}},\\vv{\\text{w}}\\bigr]=\\begin{vmatrix} \\text{u}_x &amp; \\text{u}_y &amp; \\text{u}_z \\\\[1.1ex] \\text{v}_x &amp;\\text{v}_y&amp;\\text{v}_z \\\\[1.1ex] \\text{w}_x &amp; \\text{w}_y &amp; \\text{w}_z \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"188\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<\/div>\n<p>\u6240\u4ee5\u4f60\u53ef\u4ee5\u770b\u4e00\u4e2a<span style=\"text-decoration: underline;\">\u5982\u4f55\u8ba1\u7b97\u7684\u4f8b\u5b50<\/span>\uff0c\u6211\u4eec\u5c06\u627e\u5230\u4ee5\u4e0b\u4e09\u4e2a\u5411\u91cf\u7684\u6df7\u5408\u4e58\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-d90f4dd611d93b46a9001a7fe26e03f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}= (1,2,0) \\qquad \\vv{\\text{v}}= (0,-1,3)\\qquad \\vv{\\text{w}}= (-2,4,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"369\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u4e3a\u4e86\u786e\u5b9a\u6df7\u5408\u4e58\u79ef\uff0c\u6211\u4eec\u901a\u8fc7\u5c06\u5411\u91cf\u653e\u7f6e\u5728\u77e9\u9635\u7684\u884c\u4e2d\u6765\u6784\u9020 3 \u9636\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-e5341b4a39b42c1284a4b0129b38b61a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bigl[\\vv{\\text{u}},\\vv{\\text{v}},\\vv{\\text{w}}\\bigr]=\\begin{vmatrix} 1 &amp; 2 &amp; 0 \\\\[1.1ex] 0 &amp; -1 &amp; 3 \\\\[1.1ex] -2 &amp; 4 &amp; 1 \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"180\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u73b0\u5728\u6211\u4eec\u53ea\u9700\u8981\u6c42\u89e3\u77e9\u9635\u7684\u884c\u5217\u5f0f\uff0c\u4e3a\u6b64\u4f60\u53ef\u4ee5\u4f7f\u7528\u4efb\u4f55\u65b9\u6cd5\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5c06\u5e94\u7528 Sarrus \u89c4\u5219\uff08\u4f46\u8fd9\u4e5f\u53ef\u4ee5\u901a\u8fc7\u6dfb\u52a0\u6216\u8f85\u52a9\u56e0\u5b50\u6765\u5b8c\u6210\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-df86565048cf897265878936f2294525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\bigl[\\vv{\\text{u}},\\vv{\\text{v}},\\vv{\\text{w}}\\bigr]&amp; =\\begin{vmatrix} 1 &amp; 2 &amp; 0 \\\\[1.1ex] 0 &amp; -1 &amp; 3 \\\\[1.1ex] -2 &amp; 4 &amp; 1 \\end{vmatrix} \\\\[2ex] &amp;= -1-12+0-0-12-0 \\\\[2ex] &amp; = \\bm{-25} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"166\" width=\"276\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<hr class=\"wp-block-separator has-text-color has-background has-pale-cyan-blue-background-color has-pale-cyan-blue-color is-style-wide\">\n<p>\u4e3a\u4e86\u8bc1\u660e\u8fd9\u4e24\u4e2a\u8fc7\u7a0b\u662f\u7b49\u4ef7\u7684\uff0c\u6211\u4eec\u5c06\u6839\u636e\u5b83\u4eec\u7684\u5b9a\u4e49\u6765\u8ba1\u7b97\u76f8\u540c\u5411\u91cf\u7684\u6df7\u5408\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-e4d15d610ca9b4cb39e9f268cfb152ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\bigl[\\vv{\\text{u}},\\vv{\\text{v}},\\vv{\\text{w}}\\bigr] &amp; = \\vv{\\text{u}} \\cdot ( \\vv{\\text{v}}\\times \\vv{\\text{w}})\\\\[2ex] &amp;=(1,2,0) \\cdot \\Bigl( (0,-1,3)\\times (-2,4,1)\\Bigr) \\\\[2ex] &amp; = (1,2,0) \\cdot \\begin{vmatrix} \\vv{i}&amp; \\vv{j}&amp; \\vv{k} \\\\[1.1ex] 0&amp; -1 &amp; 3 \\\\[1.1ex] -2 &amp;4&amp;1 \\end{vmatrix} \\\\[2ex] &amp;=  (1,2,0) \\cdot (-13,-6,-2) \\\\[2ex] &amp; = \\bm{-25} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"262\" width=\"332\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u6211\u4eec\u5efa\u8bae\u901a\u8fc7\u5411\u91cf\u7684\u884c\u5217\u5f0f\u6765\u8ba1\u7b97\u6df7\u5408\u4e58\u79ef\uff0c\u56e0\u4e3a\u5b83\u66f4\u5feb\u5e76\u4e14\u51fa\u9519\u7684\u673a\u4f1a\u66f4\u5c11\u3002\u4f46\u662f\uff0c\u6b63\u5982\u60a8\u6240\u770b\u5230\u7684\uff0c\u65e0\u8bba\u60a8\u4f7f\u7528\u54ea\u79cd\u65b9\u6cd5\uff0c\u7ed3\u679c\u90fd\u662f\u76f8\u540c\u7684\uff0c\u56e0\u6b64\u8bf7\u4f7f\u7528\u60a8\u559c\u6b22\u7684\u65b9\u6cd5\u3002 \ud83d\udc4d <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"interpretacion-geometrica-del-producto-mixto\"><\/span>\u6df7\u5408\u4ea7\u54c1\u7684\u51e0\u4f55\u89e3\u91ca<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u4e00\u65e6\u60a8\u77e5\u9053\u5982\u4f55\u6c42\u4e09\u4e2a\u5411\u91cf\u7684\u6df7\u5408\u79ef\uff0c\u60a8\u53ef\u80fd\u4f1a\u60f3\u2026\u2026\u6df7\u5408\u79ef\u7684\u7528\u9014\u662f\u4ec0\u4e48\uff1f\u55ef\uff0c\u5728\u6570\u5b66\u4e2d\u5b83\u6709\u4e24\u4e2a\u4e3b\u8981\u7528\u9014\uff1a\u8ba1\u7b97\u5e73\u884c\u516d\u9762\u4f53\u7684\u4f53\u79ef\u548c\u56db\u9762\u4f53\u7684\u4f53\u79ef\u3002<\/p>\n<p><strong>\u5e73\u884c\u516d\u9762\u4f53\u7684\u4f53\u79ef<\/strong>\u7b49\u4e8e\u6807\u8bb0\u51e0\u4f55\u573a\u7684 3 \u4e2a\u7ef4\u5ea6\u7684\u5411\u91cf\u7684\u6df7\u5408\u4e58\u79ef\u7684\u7edd\u5bf9\u503c\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\/exemple-de-produit-mixte-de-trois-vecteurs.webp\" alt=\"\u4e09\u4e2a\u5411\u91cf\u7684\u6df7\u5408\u4e58\u79ef\u6216\u4e09\u91cd\u6807\u91cf\u79ef\u7684\u793a\u4f8b\" class=\"wp-image-999\" width=\"309\" height=\"310\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p>\u6df7\u5408\u4e58\u79ef\u7684\u53e6\u4e00\u4e2a\u5e94\u7528\u662f\u786e\u5b9a<strong>\u56db\u9762\u4f53\u7684\u4f53\u79ef<\/strong>\u3002\u56e0\u4e3a\u5728\u51e0\u4f55\u4e0a\uff0c\u6df7\u5408\u4e58\u79ef\u7edd\u5bf9\u503c\u7684\u7b2c\u516d\u90e8\u5206\u4ee3\u8868\u56db\u9762\u4f53\u7684\u4f53\u79ef\uff1a <\/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\/produit-mixte-de-trois-vecteurs-dans-r3.webp\" alt=\"r3 \u4e2d\u4e09\u4e2a\u5411\u91cf\u7684\u6df7\u5408\u79ef\" class=\"wp-image-1002\" width=\"273\" height=\"327\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedades-del-producto-mixto-o-triple-producto-escalar\"><\/span>\u6df7\u5408\u79ef\u6216\u4e09\u70b9\u79ef\u7684\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u6df7\u5408\u4e58\u79ef\u6216\u4e09\u6807\u91cf\u4e58\u79ef\u5177\u6709\u4ee5\u4e0b\u7279\u5f81\uff1a<\/p>\n<ul>\n<li>\u4e00\u822c\u6765\u8bf4\uff0c\u6df7\u5408\u4e58\u79ef\u5411\u91cf<strong>\u987a\u5e8f<\/strong>\u7684\u53d8\u5316\u4e5f\u610f\u5473\u7740\u7b26\u53f7\u7684\u53d8\u5316\u3002\u56e0\u6b64\uff0c\u6df7\u5408\u4e58\u79ef\u5411\u91cf\u7684\u987a\u5e8f\u5f88\u91cd\u8981\u3002<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0ce0eff894cd1fa149c6d8c8ae6f0f03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bigl[\\vv{\\text{u}},\\vv{\\text{v}},\\vv{\\text{w}}\\bigr] = -\\bigl[\\vv{\\text{v}},\\vv{\\text{u}},\\vv{\\text{w}}\\bigr] = -\\bigl[\\vv{\\text{u}},\\vv{\\text{w}},\\vv{\\text{v}}\\bigr] = - \\bigl[\\vv{\\text{w}},\\vv{\\text{v}},\\vv{\\text{u}}\\bigr]\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"356\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li>\u7136\u800c\uff0c\u5982\u679c\u6211\u4eec<strong>\u5faa\u73af<\/strong>\u6539\u53d8\u987a\u5e8f\uff0c\u7b26\u53f7\u4e0d\u4f1a\u6539\u53d8\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-7e3f514d940b6d366613f59c6cd8bbba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bigl[\\vv{\\text{u}},\\vv{\\text{v}},\\vv{\\text{w}}\\bigr] = \\bigl[\\vv{\\text{v}},\\vv{\\text{w}},\\vv{\\text{u}}\\bigr] = \\bigl[\\vv{\\text{w}},\\vv{\\text{u}},\\vv{\\text{v}}\\bigr]\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"229\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li>\u5728\u4e09\u7ef4\u7a7a\u95f4\uff08R3 \u4e2d\uff09\u4e2d\uff0c\u4e09\u4e2a<strong>\u7ebf\u6027\u76f8\u5173<\/strong>\u6216<strong>\u5171\u9762\u5411<\/strong>\u91cf\uff08\u5c5e\u4e8e\u540c\u4e00\u5e73\u9762\uff09\u7684\u6df7\u5408\u79ef\u7b49\u4e8e 0\u3002 <\/li>\n<\/ul>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-productos-mixtos\"><\/span>\u4fee\u590d\u4e86\u6df7\u5408\u4ea7\u54c1\u95ee\u9898<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e601<\/h3>\n<p>\u7ed9\u5b9a 3 \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-e4ea59160804c0287d02cbd1cf01b787_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}= (3,-1,2) \\qquad \\vv{\\text{v}}= (-2,0,1)\\qquad \\vv{\\text{w}}= (5,1,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"383\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u8ba1\u7b97\u4e09\u4e2a\u5411\u91cf\u7684\u6df7\u5408\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-6f6a0d080e09991ca4dc57ff1dd1ab83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bigl[\\vv{\\text{u}},\\vv{\\text{v}},\\vv{\\text{w}}\\bigr]\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u4e3a\u4e86\u627e\u5230\u5b83\u7684\u6df7\u5408\u79ef\uff0c\u6211\u4eec\u5fc5\u987b\u6c42\u89e3\u7531\u5411\u91cf\u5750\u6807\u7ec4\u6210\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-549ed90889d541ae4a1075449567b062_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\bigl[\\vv{\\text{u}},\\vv{\\text{v}},\\vv{\\text{w}}\\bigr]&amp; =\\begin{vmatrix} 3 &amp; -1 &amp; 2 \\\\[1.1ex] -2 &amp; 0 &amp; 1 \\\\[1.1ex] 5 &amp; 1 &amp; -1 \\end{vmatrix} \\\\[2ex] &amp;= 0-5-4-0-3+2 \\\\[2ex] &amp; = \\bm{-10} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"166\" width=\"244\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e602<\/h3>\n<p>\u7ed9\u5b9a 3 \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-269e213cd1d1e52c2189fe0dc420f93b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}= (7,2,-3) \\qquad \\vv{\\text{v}}= (2,4,9)\\qquad \\vv{\\text{w}}= (4,3,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"369\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u786e\u5b9a\u4e09\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u6df7\u5408\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-6f6a0d080e09991ca4dc57ff1dd1ab83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bigl[\\vv{\\text{u}},\\vv{\\text{v}},\\vv{\\text{w}}\\bigr]\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u4e3a\u4e86\u627e\u5230\u5b83\u7684\u6df7\u5408\u4e58\u79ef\uff0c\u6211\u4eec\u9700\u8981\u6c42\u89e3\u5177\u6709\u7ebf\u6027\u5f62\u5f0f\u5411\u91cf\u7684\u7b1b\u5361\u5c14\u5750\u6807\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-29b46155243552a99f6fd75de69f59da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\bigl[\\vv{\\text{u}},\\vv{\\text{v}},\\vv{\\text{w}}\\bigr]&amp; =\\begin{vmatrix} 7 &amp; 2 &amp; -3 \\\\[1.1ex] 2 &amp; 4 &amp; 9 \\\\[1.1ex] 4 &amp; 3 &amp; -1 \\end{vmatrix} \\\\[2ex] &amp;= -28+72-18+48-189+4 \\\\[2ex] &amp; = \\bm{-111} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"166\" width=\"312\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e603<\/h3>\n<p>\u8ba1\u7b97 3 \u6761\u8fb9\u4e3a\u4ee5\u4e0b\u5411\u91cf\u7684\u5e73\u884c\u516d\u9762\u4f53\u7684\u4f53\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-897316aad94e2c06a745cc34bdcd2cb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}= (0,2,5) \\qquad \\vv{\\text{v}}= (-1,6,2)\\qquad \\vv{\\text{w}}= (3,1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"355\" 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\u884c\u516d\u9762\u4f53\u7684\u4f53\u79ef\u7b49\u4e8e\u5176\u4f5c\u4e3a\u8fb9\u7684 3 \u4e2a\u5411\u91cf\u7684\u6df7\u5408\u4e58\u79ef\u7684\u7edd\u5bf9\u503c\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9996\u5148\u8ba1\u7b97\u5411\u91cf\u7684\u4e09\u91cd\u53c9\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-9718b369a508216fbb69ab68ac7de381_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\bigl[\\vv{\\text{u}},\\vv{\\text{v}},\\vv{\\text{w}}\\bigr]&amp; =\\begin{vmatrix} 0 &amp; 2 &amp; 5 \\\\[1.1ex] -1 &amp; 6 &amp; 2 \\\\[1.1ex] 3 &amp; 1 &amp; 2 \\end{vmatrix} \\\\[2ex] &amp;= 0+12-5-90-0+4 \\\\[2ex] &amp; = -79 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"166\" width=\"263\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u8fd9\u6837\u5e73\u884c\u516d\u9762\u4f53\u7684\u4f53\u79ef\u5c31\u662f\u6df7\u5408\u4e58\u79ef\u7ed3\u679c\u7684\u7edd\u5bf9\u503c\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-6074917d7c095bd718364a5ed3081c2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"V= \\lvert -79 \\rvert = \\bm{79}\\ \\mathbf{u}\\bm{^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"145\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e604<\/h3>\n<p>\u8ba1\u7b97\u9876\u70b9\u4e3a\u4ee5\u4e0b\u70b9\u7684\u56db\u9762\u4f53\u7684\u4f53\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-4c798eaa40f98c461a076a2bc0aa0910_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(1,0,2) \\qquad B(3,3,2)\\qquad C(5,-1,4)\\qquad D(4,2,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"401\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u9996\u5148\uff0c\u6211\u4eec\u8ba1\u7b97\u4ee3\u8868\u56db\u9762\u4f53\u8fb9\u7f18\u7684\u5411\u91cf\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fe16cd346a46022ffcd90ad080b305ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB}=B-A= (3,3,2)-(1,0,2)=(2,3,0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"337\" 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-7bb115ce61905a3f3a5c76dfd152b8f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AC}=C-A= (5,-1,4)-(1,0,2)=(4,-1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"364\" 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-11697d137db244fd5b7a5f6328181183_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AD}=D-A= (4,2,1)-(1,0,2)=(3,2,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"353\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56db\u9762\u4f53\u7684\u4f53\u79ef\u7b49\u4e8e\u5176 3 \u4e2a\u8fb9\u5411\u91cf\u7684\u6df7\u5408\u4e58\u79ef\u7edd\u5bf9\u503c\u7684\u516d\u5206\u4e4b\u4e00\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9996\u5148\u8ba1\u7b97\u627e\u5230\u7684\u5411\u91cf\u7684\u6df7\u5408\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-d384e76415a756b834e0a8e7c695b1c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\bigl[\\vv{\\text{u}},\\vv{\\text{v}},\\vv{\\text{w}}\\bigr]&amp; =\\begin{vmatrix} 2 &amp; 3 &amp; 0 \\\\[1.1ex] 4 &amp; -1 &amp; 2 \\\\[1.1ex] 3 &amp; 2 &amp; -1 \\end{vmatrix} \\\\[2ex] &amp;= 2+18+0-0-8+12 \\\\[2ex] &amp; = 24 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"166\" width=\"262\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u56e0\u6b64\uff0c\u56db\u9762\u4f53\u7684\u4f53\u79ef\u5c06\u662f\u6df7\u5408\u4e58\u79ef\u7edd\u5bf9\u503c\u7684\u516d\u5206\u4e4b\u4e00\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-0e1aefc732626b5a6d55a904d03452b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"V= \\cfrac{1}{6} \\cdot \\lvert 24 \\rvert = \\cfrac{24}{6} = \\bm{4} \\ \\mathbf{u}\\bm{^3}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"189\" 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\u6211\u4eec\u89e3\u91ca\u4ec0\u4e48\u662f\u4e09\u4e2a\u5411\u91cf\u7684\u6df7\u5408\u79ef\uff08\u6216\u4e09\u91cd\u70b9\u79ef\uff09\u4ee5\u53ca\u5b83\u7684\u8ba1\u7b97\u65b9\u6cd5\u3002\u60a8\u8fd8\u5c06\u770b\u5230\u6709\u5173\u5411\u91cf\u4e4b\u95f4\u6b64\u7c7b\u8fd0\u7b97\u7684\u793a\u4f8b &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/cn\/%e4%b8%89%e4%b8%aa%e5%90%91%e9%87%8f%e6%88%96%e4%b8%89%e9%87%8d%e6%a0%87%e9%87%8f%e7%a7%af%e7%9a%84%e6%b7%b7%e5%90%88%e7%a7%af%e7%a4%ba%e4%be%8b\/\"> <span class=\"screen-reader-text\">\u4e09\u4e2a\u5411\u91cf\u7684\u6df7\u5408\u79ef\uff08\u6216\u4e09\u91cd\u70b9\u79ef\uff09<\/span> \u67e5\u770b\u5168\u6587 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