{"id":11089,"date":"2026-09-21T07:12:23","date_gmt":"2026-09-21T07:12:23","guid":{"rendered":"https:\/\/otus.ru\/journal\/?p=11089"},"modified":"2026-09-25T15:39:52","modified_gmt":"2026-09-25T15:39:52","slug":"vse-o-vektorah-ot-a-do-ya","status":"publish","type":"post","link":"https:\/\/otus.ru\/journal\/vse-o-vektorah-ot-a-do-ya\/","title":{"rendered":"\u0412\u0435\u043a\u0442\u043e\u0440\u044b: \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0438, \u0434\u043b\u0438\u043d\u0430 \u0438 \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435 \u0432 ML \u0438 \u0433\u0440\u0430\u0444\u0438\u043a\u0435"},"content":{"rendered":"<p><strong>\u0412\u0435\u043a\u0442\u043e\u0440<\/strong> &#8212; \u044d\u0442\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u043e\u0431\u044a\u0435\u043a\u0442, \u0443 \u043a\u043e\u0442\u043e\u0440\u043e\u0433\u043e \u0435\u0441\u0442\u044c \u0434\u043b\u0438\u043d\u0430 \u0438 \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435. \u0423 \u043d\u0435\u0433\u043e \u0434\u0432\u0430 \u0440\u0430\u0432\u043d\u043e\u043f\u0440\u0430\u0432\u043d\u044b\u0445 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u0438\u044f, \u0438 \u0438\u0445 \u0432\u0430\u0436\u043d\u043e \u043d\u0435 \u043f\u0443\u0442\u0430\u0442\u044c. \u0413\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438 \u0432\u0435\u043a\u0442\u043e\u0440 &#8212; \u044d\u0442\u043e \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u043d\u044b\u0439 \u043e\u0442\u0440\u0435\u0437\u043e\u043a (\u0441\u0442\u0440\u0435\u043b\u043a\u0430) \u0438\u0437 \u0442\u043e\u0447\u043a\u0438 A \u0432 \u0442\u043e\u0447\u043a\u0443 B. \u0410\u043b\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043a\u0438 \u0442\u043e\u0442 \u0436\u0435 \u0432\u0435\u043a\u0442\u043e\u0440 &#8212; \u044d\u0442\u043e \u0443\u043f\u043e\u0440\u044f\u0434\u043e\u0447\u0435\u043d\u043d\u044b\u0439 \u043d\u0430\u0431\u043e\u0440 \u0447\u0438\u0441\u0435\u043b (\u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442), \u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">[3, 4]<\/code>. \u041e\u0434\u043d\u0430 \u0438 \u0442\u0430 \u0436\u0435 \u0441\u0442\u0440\u0435\u043b\u043a\u0430 \u0438\u0437 \u043d\u0430\u0447\u0430\u043b\u0430 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442 \u0432 \u0442\u043e\u0447\u043a\u0443 (3, 4) \u0437\u0430\u043f\u0438\u0441\u044b\u0432\u0430\u0435\u0442\u0441\u044f \u043a\u0430\u043a \u043d\u0430\u0431\u043e\u0440 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">[3, 4]<\/code>, \u0438 \u0435\u0435 \u0434\u043b\u0438\u043d\u0430 \u0440\u0430\u0432\u043d\u0430 5.<\/p>\n<div class=\"oj-toc\" style=\"background:#f6f6f6;border-radius:8px;padding:14px 20px;margin:20px 0\">\n<p><strong>\u0421\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u0435<\/strong><\/p>\n<ol>\n<li><a href=\"#s1\">\u0414\u0432\u0430 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u043e\u0434\u043d\u043e\u0433\u043e \u0432\u0435\u043a\u0442\u043e\u0440\u0430<\/a><\/li>\n<li><a href=\"#s2\">\u0414\u043b\u0438\u043d\u0430 (\u043d\u043e\u0440\u043c\u0430) \u0432\u0435\u043a\u0442\u043e\u0440\u0430<\/a><\/li>\n<li><a href=\"#s3\">\u0421\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0438 \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u0441\u043a\u0430\u043b\u044f\u0440<\/a><\/li>\n<li><a href=\"#s4\">\u0421\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435<\/a><\/li>\n<li><a href=\"#s5\">\u0423\u0433\u043e\u043b \u043c\u0435\u0436\u0434\u0443 \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438 \u0447\u0435\u0440\u0435\u0437 \u043a\u043e\u0441\u0438\u043d\u0443\u0441<\/a><\/li>\n<li><a href=\"#s6\">\u041d\u043e\u0440\u043c\u0430\u043b\u0438\u0437\u0430\u0446\u0438\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430<\/a><\/li>\n<li><a href=\"#s7\">\u0412\u0435\u043a\u0442\u043e\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 (\u0442\u043e\u043b\u044c\u043a\u043e \u0432 3D)<\/a><\/li>\n<li><a href=\"#s8\">\u0413\u0434\u0435 \u0432\u0435\u043a\u0442\u043e\u0440\u044b \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u044e\u0442\u0441\u044f<\/a><\/li>\n<li><a href=\"#s9\">\u0412\u044b\u0432\u043e\u0434\u044b<\/a><\/li>\n<li><a href=\"#s10\">\u0413\u0434\u0435 \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u0435\u0442\u0441\u044f \/ \u0441\u0432\u044f\u0437\u044c \u0441 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u043e\u0439<\/a><\/li>\n<li><a href=\"#s11\">FAQ<\/a><\/li>\n<\/ol>\n<\/div>\n<p>\u041d\u0438\u0436\u0435 \u0440\u0430\u0437\u0431\u0435\u0440\u0443 \u043e\u0431\u0435 \u0442\u0440\u0430\u043a\u0442\u043e\u0432\u043a\u0438, \u043e\u0441\u043d\u043e\u0432\u043d\u044b\u0435 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0438 \u043d\u0430\u0434 \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438 (\u0441\u043b\u043e\u0436\u0435\u043d\u0438\u0435, \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u0441\u043a\u0430\u043b\u044f\u0440, \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u0438 \u0432\u0435\u043a\u0442\u043e\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435, \u043d\u043e\u0440\u043c\u0443, \u043d\u043e\u0440\u043c\u0430\u043b\u0438\u0437\u0430\u0446\u0438\u044e, \u0443\u0433\u043e\u043b \u0447\u0435\u0440\u0435\u0437 \u043a\u043e\u0441\u0438\u043d\u0443\u0441) \u0441 \u0444\u043e\u0440\u043c\u0443\u043b\u0430\u043c\u0438 \u0438 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u043c\u0438, \u0430 \u0437\u0430\u0442\u0435\u043c \u043f\u043e\u043a\u0430\u0436\u0443, \u0433\u0434\u0435 \u044d\u0442\u043e \u0440\u0430\u0431\u043e\u0442\u0430\u0435\u0442: \u043c\u0430\u0448\u0438\u043d\u043d\u043e\u0435 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435, \u043a\u043e\u043c\u043f\u044c\u044e\u0442\u0435\u0440\u043d\u0430\u044f \u0433\u0440\u0430\u0444\u0438\u043a\u0430, \u0444\u0438\u0437\u0438\u043a\u0430. \u041f\u0440\u0438\u043c\u0435\u0440\u044b &#8212; \u043d\u0430 Python \u0441 \u0431\u0438\u0431\u043b\u0438\u043e\u0442\u0435\u043a\u043e\u0439 numpy, \u0441 \u043f\u0440\u043e\u0433\u043e\u0432\u043e\u0440\u0435\u043d\u043d\u044b\u043c \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u043e\u043c.<\/p>\n<h2 id=\"s1\">\u0414\u0432\u0430 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u043e\u0434\u043d\u043e\u0433\u043e \u0432\u0435\u043a\u0442\u043e\u0440\u0430<\/h2>\n<p>\u0427\u0442\u043e\u0431\u044b \u0434\u0430\u043b\u044c\u0448\u0435 \u043d\u0435 \u043f\u0443\u0442\u0430\u0442\u044c\u0441\u044f, \u0437\u0430\u0444\u0438\u043a\u0441\u0438\u0440\u0443\u044e \u043c\u0438\u043d\u0438-\u0446\u0435\u043f\u043e\u0447\u043a\u0443 \u00ab\u043e\u0431\u044a\u0435\u043a\u0442 -&gt; \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u044b -&gt; \u0447\u0438\u0441\u043b\u043e\u00bb:<\/p>\n<ul>\n<li><strong>\u0413\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0432\u0435\u043a\u0442\u043e\u0440<\/strong> &#8212; \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u043d\u044b\u0439 \u043e\u0442\u0440\u0435\u0437\u043e\u043a. \u0415\u0441\u0442\u044c \u043d\u0430\u0447\u0430\u043b\u043e, \u043a\u043e\u043d\u0435\u0446, \u0434\u043b\u0438\u043d\u0430 \u0438 \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435. \u0421\u0442\u0440\u0435\u043b\u043a\u0443 \u043c\u043e\u0436\u043d\u043e \u0434\u0432\u0438\u0433\u0430\u0442\u044c: \u0435\u0441\u043b\u0438 \u0434\u043b\u0438\u043d\u0430 \u0438 \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u0442\u0435 \u0436\u0435, \u044d\u0442\u043e \u0442\u043e\u0442 \u0436\u0435 (\u0440\u0430\u0432\u043d\u044b\u0439) \u0432\u0435\u043a\u0442\u043e\u0440.<\/li>\n<li><strong>\u041a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u044b \u0432\u0435\u043a\u0442\u043e\u0440\u0430<\/strong> &#8212; \u0443\u043f\u043e\u0440\u044f\u0434\u043e\u0447\u0435\u043d\u043d\u044b\u0439 \u043d\u0430\u0431\u043e\u0440 \u0447\u0438\u0441\u0435\u043b. \u0421\u0442\u0440\u0435\u043b\u043a\u0443 \u0438\u0437 \u043d\u0430\u0447\u0430\u043b\u0430 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">[0, 0]<\/code> \u0432 \u0442\u043e\u0447\u043a\u0443 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">(3, 4)<\/code> \u043e\u043f\u0438\u0441\u044b\u0432\u0430\u044e\u0442 \u0434\u0432\u0430 \u0447\u0438\u0441\u043b\u0430: \u0441\u0434\u0432\u0438\u0433 \u043f\u043e \u043e\u0441\u0438 X \u0438 \u043f\u043e \u043e\u0441\u0438 Y. \u041f\u043e\u0440\u044f\u0434\u043e\u043a \u0432\u0430\u0436\u0435\u043d: <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">[3, 4]<\/code> \u0438 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">[4, 3]<\/code> &#8212; \u0440\u0430\u0437\u043d\u044b\u0435 \u0432\u0435\u043a\u0442\u043e\u0440\u044b.<\/li>\n<li><strong>\u0421\u043a\u0430\u043b\u044f\u0440<\/strong> &#8212; \u044d\u0442\u043e \u043f\u0440\u043e\u0441\u0442\u043e \u0447\u0438\u0441\u043b\u043e (\u0431\u0435\u0437 \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u044f): \u0434\u043b\u0438\u043d\u0430 \u0432\u0435\u043a\u0442\u043e\u0440\u0430 \u0438\u043b\u0438 \u043c\u043d\u043e\u0436\u0438\u0442\u0435\u043b\u044c. \u041f\u0443\u0442\u0430\u0442\u044c \u0441\u043e \u0441\u043a\u0430\u043b\u044f\u0440\u043e\u043c \u043d\u0435\u043b\u044c\u0437\u044f: <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">2<\/code> &#8212; \u0441\u043a\u0430\u043b\u044f\u0440, <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">[2, 0]<\/code> &#8212; \u0432\u0435\u043a\u0442\u043e\u0440.<\/li>\n<\/ul>\n<p>\u0427\u0438\u0441\u043b\u043e \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442 \u0437\u0430\u0434\u0430\u0435\u0442 \u0440\u0430\u0437\u043c\u0435\u0440\u043d\u043e\u0441\u0442\u044c: \u0434\u0432\u0430 \u0447\u0438\u0441\u043b\u0430 &#8212; \u0434\u0432\u0443\u043c\u0435\u0440\u043d\u044b\u0439 \u0432\u0435\u043a\u0442\u043e\u0440 \u043d\u0430 \u043f\u043b\u043e\u0441\u043a\u043e\u0441\u0442\u0438 XY, \u0442\u0440\u0438 &#8212; \u0442\u0440\u0435\u0445\u043c\u0435\u0440\u043d\u044b\u0439 \u0432 \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0435 XYZ, \u0431\u043e\u043b\u044c\u0448\u0435 \u0442\u0440\u0435\u0445 &#8212; \u043c\u043d\u043e\u0433\u043e\u043c\u0435\u0440\u043d\u044b\u0439. \u041c\u043d\u043e\u0433\u043e\u043c\u0435\u0440\u043d\u044b\u0435 \u0432\u0435\u043a\u0442\u043e\u0440\u044b \u0433\u0440\u0430\u0444\u0438\u0447\u0435\u0441\u043a\u0438 \u043d\u0435 \u043d\u0430\u0440\u0438\u0441\u043e\u0432\u0430\u0442\u044c, \u043d\u043e \u0430\u043b\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043a\u0438 \u0441 \u043d\u0438\u043c\u0438 \u0440\u0430\u0431\u043e\u0442\u0430\u044e\u0442 \u0442\u0430\u043a \u0436\u0435 &#8212; \u0438\u043c\u0435\u043d\u043d\u043e \u043e\u043d\u0438 \u0447\u0430\u0449\u0435 \u0432\u0441\u0435\u0433\u043e \u0432\u0441\u0442\u0440\u0435\u0447\u0430\u044e\u0442\u0441\u044f \u0432 \u043c\u0430\u0448\u0438\u043d\u043d\u043e\u043c \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0438.<\/p>\n<h2 id=\"s2\">\u0414\u043b\u0438\u043d\u0430 (\u043d\u043e\u0440\u043c\u0430) \u0432\u0435\u043a\u0442\u043e\u0440\u0430<\/h2>\n<p><strong>\u0414\u043b\u0438\u043d\u0430 \u0432\u0435\u043a\u0442\u043e\u0440\u0430<\/strong> (\u043c\u043e\u0434\u0443\u043b\u044c, \u0438\u043b\u0438 \u043d\u043e\u0440\u043c\u0430) &#8212; \u044d\u0442\u043e \u043d\u0435\u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0435 \u0447\u0438\u0441\u043b\u043e, \u043f\u043e\u043a\u0430\u0437\u044b\u0432\u0430\u044e\u0449\u0435\u0435 \u00ab\u0432\u0435\u043b\u0438\u0447\u0438\u043d\u0443\u00bb \u0441\u0442\u0440\u0435\u043b\u043a\u0438. \u041e\u0431\u043e\u0437\u043d\u0430\u0447\u0430\u0435\u0442\u0441\u044f \u043a\u0430\u043a <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">|a|<\/code> \u0438\u043b\u0438 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">||a||<\/code>. \u0414\u043b\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430 \u0441 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u0430\u043c\u0438 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">[x, y]<\/code> \u0434\u043b\u0438\u043d\u0430 \u0441\u0447\u0438\u0442\u0430\u0435\u0442\u0441\u044f \u043f\u043e \u0442\u0435\u043e\u0440\u0435\u043c\u0435 \u041f\u0438\u0444\u0430\u0433\u043e\u0440\u0430:<\/p>\n<p>|a| = sqrt(x^2 + y^2)<\/p>\n<p>\u0414\u043b\u044f \u0442\u0440\u0435\u0445\u043c\u0435\u0440\u043d\u043e\u0433\u043e \u0432\u0435\u043a\u0442\u043e\u0440\u0430 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">[x, y, z]<\/code> \u043f\u043e\u0434 \u043a\u043e\u0440\u043d\u0435\u043c \u0441\u0442\u043e\u0438\u0442 \u0441\u0443\u043c\u043c\u0430 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043e\u0432 \u0432\u0441\u0435\u0445 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442. \u042d\u0442\u043e \u0435\u0432\u043a\u043b\u0438\u0434\u043e\u0432\u0430 \u043d\u043e\u0440\u043c\u0430 (L2); \u0435\u0441\u0442\u044c \u0438 \u0434\u0440\u0443\u0433\u0438\u0435 \u043d\u043e\u0440\u043c\u044b (\u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440 L1 &#8212; \u0441\u0443\u043c\u043c\u0430 \u043c\u043e\u0434\u0443\u043b\u0435\u0439 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442), \u043d\u043e \u043f\u043e \u0443\u043c\u043e\u043b\u0447\u0430\u043d\u0438\u044e \u043f\u043e\u0434 \u00ab\u0434\u043b\u0438\u043d\u043e\u0439\u00bb \u043f\u043e\u043d\u0438\u043c\u0430\u044e\u0442 \u0438\u043c\u0435\u043d\u043d\u043e L2.<\/p>\n<p>\u041c\u0438\u043d\u0438\u043c\u0430\u043b\u044c\u043d\u044b\u0439 \u0440\u0430\u0431\u043e\u0447\u0438\u0439 \u043f\u0440\u0438\u043c\u0435\u0440 \u043d\u0430 numpy:<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\"><span style=\"color: #C678DD\">import<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">numpy<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #C678DD\">as<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">np<\/span>\n\n<span style=\"color: #E06C75\">a<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">array<\/span><span style=\"color: #ABB2BF\">([<\/span><span style=\"color: #D19A66\">3<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">4<\/span><span style=\"color: #ABB2BF\">])<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">linalg<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">norm<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">a<\/span><span style=\"color: #ABB2BF\">))<\/span>  <span style=\"color: #7F848E\"># 5.0<\/span><\/code><\/pre>\n<p>\u0412 \u043a\u043e\u043d\u0441\u043e\u043b\u0438 \u043f\u043e\u044f\u0432\u0438\u0442\u0441\u044f <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">5.0<\/code>: sqrt(3^2 + 4^2) = sqrt(25) = 5. \u0414\u043b\u0438\u043d\u0430 \u0432\u0441\u0435\u0433\u0434\u0430 \u043d\u0435\u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u0430 \u0438 \u0440\u0430\u0432\u043d\u0430 \u043d\u0443\u043b\u044e \u0442\u043e\u043b\u044c\u043a\u043e \u0443 \u043d\u0443\u043b\u0435\u0432\u043e\u0433\u043e \u0432\u0435\u043a\u0442\u043e\u0440\u0430 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">[0, 0]<\/code>.<\/p>\n<h2 id=\"s3\">\u0421\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0438 \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u0441\u043a\u0430\u043b\u044f\u0440<\/h2>\n<p>\u041e\u0431\u0435 \u0431\u0430\u0437\u043e\u0432\u044b\u0435 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0438 \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u044e\u0442\u0441\u044f \u043f\u043e\u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u043d\u043e.<\/p>\n<p><strong>\u0421\u043b\u043e\u0436\u0435\u043d\u0438\u0435.<\/strong> \u041a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u044b \u0441\u043a\u043b\u0430\u0434\u044b\u0432\u0430\u044e\u0442 \u043f\u043e\u043f\u0430\u0440\u043d\u043e: <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">[1, 2] + [3, 4] = [4, 6]<\/code>. \u0413\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438 \u044d\u0442\u043e \u00ab\u043f\u0440\u0430\u0432\u0438\u043b\u043e \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0430\u00bb: \u043f\u0440\u0438\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u043c \u043d\u0430\u0447\u0430\u043b\u043e \u0432\u0442\u043e\u0440\u043e\u0439 \u0441\u0442\u0440\u0435\u043b\u043a\u0438 \u043a \u043a\u043e\u043d\u0446\u0443 \u043f\u0435\u0440\u0432\u043e\u0439, \u0430 \u0441\u0443\u043c\u043c\u0430 &#8212; \u0441\u0442\u0440\u0435\u043b\u043a\u0430 \u043e\u0442 \u043d\u0430\u0447\u0430\u043b\u0430 \u043f\u0435\u0440\u0432\u043e\u0439 \u043a \u043a\u043e\u043d\u0446\u0443 \u0432\u0442\u043e\u0440\u043e\u0439. \u0421\u043a\u043b\u0430\u0434\u044b\u0432\u0430\u0442\u044c \u043c\u043e\u0436\u043d\u043e \u0442\u043e\u043b\u044c\u043a\u043e \u0432\u0435\u043a\u0442\u043e\u0440\u044b \u043e\u0434\u0438\u043d\u0430\u043a\u043e\u0432\u043e\u0439 \u0440\u0430\u0437\u043c\u0435\u0440\u043d\u043e\u0441\u0442\u0438.<\/p>\n<p><strong>\u0423\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u0441\u043a\u0430\u043b\u044f\u0440.<\/strong> \u041a\u0430\u0436\u0434\u0443\u044e \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u0443 \u0443\u043c\u043d\u043e\u0436\u0430\u0435\u043c \u043d\u0430 \u0447\u0438\u0441\u043b\u043e <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">k<\/code>: <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">2 * [1, 2] = [2, 4]<\/code>. \u0427\u0442\u043e \u043f\u0440\u043e\u0438\u0441\u0445\u043e\u0434\u0438\u0442 \u0441 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u043c:<\/p>\n<table>\n<thead>\n<tr>\n<th>\u0417\u043d\u0430\u0447\u0435\u043d\u0438\u0435 k<\/th>\n<th>\u0427\u0442\u043e \u043f\u0440\u043e\u0438\u0441\u0445\u043e\u0434\u0438\u0442 \u0441 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u043c<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>k &gt; 1<\/td>\n<td>\u0434\u043b\u0438\u043d\u0430 \u0440\u0430\u0441\u0442\u0435\u0442 \u0432 k \u0440\u0430\u0437, \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u0442\u043e \u0436\u0435<\/td>\n<\/tr>\n<tr>\n<td>0 &lt; k &lt; 1<\/td>\n<td>\u0434\u043b\u0438\u043d\u0430 \u0443\u043c\u0435\u043d\u044c\u0448\u0430\u0435\u0442\u0441\u044f, \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u0442\u043e \u0436\u0435<\/td>\n<\/tr>\n<tr>\n<td>k = 1<\/td>\n<td>\u0432\u0435\u043a\u0442\u043e\u0440 \u043d\u0435 \u043c\u0435\u043d\u044f\u0435\u0442\u0441\u044f<\/td>\n<\/tr>\n<tr>\n<td>k &lt; 0<\/td>\n<td>\u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u0440\u0430\u0437\u0432\u043e\u0440\u0430\u0447\u0438\u0432\u0430\u0435\u0442\u0441\u044f \u043d\u0430 \u043f\u0440\u043e\u0442\u0438\u0432\u043e\u043f\u043e\u043b\u043e\u0436\u043d\u043e\u0435<\/td>\n<\/tr>\n<tr>\n<td>k = 0<\/td>\n<td>\u043f\u043e\u043b\u0443\u0447\u0430\u0435\u0442\u0441\u044f \u043d\u0443\u043b\u0435\u0432\u043e\u0439 \u0432\u0435\u043a\u0442\u043e\u0440 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">[0, 0]<\/code><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u041f\u0440\u0438\u043c\u0435\u0440 \u043d\u0430 numpy \u0441\u0440\u0430\u0437\u0443 \u0441 \u043d\u0435\u0441\u043a\u043e\u043b\u044c\u043a\u0438\u043c\u0438 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044f\u043c\u0438:<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\"><span style=\"color: #C678DD\">import<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">numpy<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #C678DD\">as<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">np<\/span>\n\n<span style=\"color: #E06C75\">a<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">array<\/span><span style=\"color: #ABB2BF\">([<\/span><span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">2<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">3<\/span><span style=\"color: #ABB2BF\">])<\/span>\n<span style=\"color: #E06C75\">b<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">array<\/span><span style=\"color: #ABB2BF\">([<\/span><span style=\"color: #D19A66\">4<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">5<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">6<\/span><span style=\"color: #ABB2BF\">])<\/span>\n\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">a<\/span> <span style=\"color: #56B6C2\">+<\/span> <span style=\"color: #E06C75\">b<\/span><span style=\"color: #ABB2BF\">)<\/span>   <span style=\"color: #7F848E\"># [5 7 9]<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #D19A66\">2<\/span> <span style=\"color: #56B6C2\">*<\/span> <span style=\"color: #E06C75\">a<\/span><span style=\"color: #ABB2BF\">)<\/span>   <span style=\"color: #7F848E\"># [2 4 6]<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">a<\/span> <span style=\"color: #56B6C2\">-<\/span> <span style=\"color: #E06C75\">b<\/span><span style=\"color: #ABB2BF\">)<\/span>   <span style=\"color: #7F848E\"># [-3 -3 -3]<\/span><\/code><\/pre>\n<p>\u0412\u044b\u0432\u043e\u0434: <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">[5 7 9]<\/code> \u0434\u043b\u044f \u0441\u0443\u043c\u043c\u044b, <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">[2 4 6]<\/code> \u0434\u043b\u044f \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u044f \u043d\u0430 2, <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">[-3 -3 -3]<\/code> \u0434\u043b\u044f \u0440\u0430\u0437\u043d\u043e\u0441\u0442\u0438. \u0412\u044b\u0447\u0438\u0442\u0430\u043d\u0438\u0435 &#8212; \u043d\u0435 \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u0430\u044f \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044f, \u0430 \u0441\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0441 \u043f\u0440\u043e\u0442\u0438\u0432\u043e\u043f\u043e\u043b\u043e\u0436\u043d\u044b\u043c \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u043c: <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">a - b = a + (-1) * b<\/code>.<\/p>\n<h2 id=\"s4\">\u0421\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435<\/h2>\n<p><strong>\u0421\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435<\/strong> (dot product) \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432 \u0434\u0430\u0435\u0442 \u0447\u0438\u0441\u043b\u043e (\u0441\u043a\u0430\u043b\u044f\u0440), \u0430 \u043d\u0435 \u0432\u0435\u043a\u0442\u043e\u0440 &#8212; \u043e\u0442\u0441\u044e\u0434\u0430 \u0438 \u043d\u0430\u0437\u0432\u0430\u043d\u0438\u0435. \u0421\u0447\u0438\u0442\u0430\u0435\u0442\u0441\u044f \u043a\u0430\u043a \u0441\u0443\u043c\u043c\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0439 \u0441\u043e\u043e\u0442\u0432\u0435\u0442\u0441\u0442\u0432\u0443\u044e\u0449\u0438\u0445 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442:<\/p>\n<p>a . b = x1<em>x2 + y1<\/em>y2 + z1*z2<\/p>\n<p>\u0415\u0441\u0442\u044c \u0432\u0442\u043e\u0440\u0430\u044f, \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0444\u043e\u0440\u043c\u0443\u043b\u0430, \u0441\u0432\u044f\u0437\u044b\u0432\u0430\u044e\u0449\u0430\u044f \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0441 \u0443\u0433\u043b\u043e\u043c:<\/p>\n<p>a . b = |a| * |b| * cos(theta)<\/p>\n<p>\u0433\u0434\u0435 theta &#8212; \u0443\u0433\u043e\u043b \u043c\u0435\u0436\u0434\u0443 \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438 (\u0444\u043e\u0440\u043c\u0443\u043b\u0430 \u0432\u0435\u0440\u043d\u0430 \u0434\u043b\u044f \u043d\u0435\u043d\u0443\u043b\u0435\u0432\u044b\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432). \u041e\u0442\u0441\u044e\u0434\u0430 \u0432\u0430\u0436\u043d\u043e\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u043e: \u0435\u0441\u043b\u0438 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0440\u0430\u0432\u043d\u043e \u043d\u0443\u043b\u044e, \u0442\u043e cos(theta) = 0, \u0442\u043e \u0435\u0441\u0442\u044c \u0432\u0435\u043a\u0442\u043e\u0440\u044b \u043f\u0435\u0440\u043f\u0435\u043d\u0434\u0438\u043a\u0443\u043b\u044f\u0440\u043d\u044b (\u0443\u0433\u043e\u043b 90 \u0433\u0440\u0430\u0434\u0443\u0441\u043e\u0432).<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\"><span style=\"color: #C678DD\">import<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">numpy<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #C678DD\">as<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">np<\/span>\n\n<span style=\"color: #E06C75\">a<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">array<\/span><span style=\"color: #ABB2BF\">([<\/span><span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">2<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">3<\/span><span style=\"color: #ABB2BF\">])<\/span>\n<span style=\"color: #E06C75\">b<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">array<\/span><span style=\"color: #ABB2BF\">([<\/span><span style=\"color: #D19A66\">4<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">5<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">6<\/span><span style=\"color: #ABB2BF\">])<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">dot<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">a<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">b<\/span><span style=\"color: #ABB2BF\">))<\/span>  <span style=\"color: #7F848E\"># 32<\/span><\/code><\/pre>\n<p>\u0420\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">32<\/code>: 1<em>4 + 2<\/em>5 + 3*6 = 4 + 10 + 18 = 32. \u0421\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0440\u0430\u0431\u043e\u0442\u0430\u0435\u0442 \u0434\u043b\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432 \u043b\u044e\u0431\u043e\u0439 \u0440\u0430\u0437\u043c\u0435\u0440\u043d\u043e\u0441\u0442\u0438 &#8212; \u0432 \u044d\u0442\u043e\u043c \u0435\u0433\u043e \u043e\u0442\u043b\u0438\u0447\u0438\u0435 \u043e\u0442 \u0432\u0435\u043a\u0442\u043e\u0440\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u043d\u0438\u0436\u0435.<\/p>\n<h2 id=\"s5\">\u0423\u0433\u043e\u043b \u043c\u0435\u0436\u0434\u0443 \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438 \u0447\u0435\u0440\u0435\u0437 \u043a\u043e\u0441\u0438\u043d\u0443\u0441<\/h2>\n<p>\u0418\u0437 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0444\u043e\u0440\u043c\u0443\u043b\u044b \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u043b\u0435\u0433\u043a\u043e \u0432\u044b\u0440\u0430\u0437\u0438\u0442\u044c \u043a\u043e\u0441\u0438\u043d\u0443\u0441 \u0443\u0433\u043b\u0430:<\/p>\n<p>cos(theta) = (a . b) \/ (|a| * |b|)<\/p>\n<p>\u0424\u043e\u0440\u043c\u0443\u043b\u0430 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0430 \u0442\u043e\u043b\u044c\u043a\u043e \u0434\u043b\u044f \u0434\u0432\u0443\u0445 \u043d\u0435\u043d\u0443\u043b\u0435\u0432\u044b\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432 (\u0438\u043d\u0430\u0447\u0435 \u0432 \u0437\u043d\u0430\u043c\u0435\u043d\u0430\u0442\u0435\u043b\u0435 \u043d\u043e\u043b\u044c). \u0417\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u043a\u043e\u0441\u0438\u043d\u0443\u0441\u0430 \u043b\u0435\u0436\u0438\u0442 \u0432 \u0434\u0438\u0430\u043f\u0430\u0437\u043e\u043d\u0435 \u043e\u0442 -1 \u0434\u043e 1: <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">1<\/code> &#8212; \u0432\u0435\u043a\u0442\u043e\u0440\u044b \u0441\u043e\u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u044b (\u0443\u0433\u043e\u043b 0), <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">0<\/code> &#8212; \u043f\u0435\u0440\u043f\u0435\u043d\u0434\u0438\u043a\u0443\u043b\u044f\u0440\u043d\u044b (90 \u0433\u0440\u0430\u0434\u0443\u0441\u043e\u0432), <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">-1<\/code> &#8212; \u043f\u0440\u043e\u0442\u0438\u0432\u043e\u043f\u043e\u043b\u043e\u0436\u043d\u043e \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u044b (180 \u0433\u0440\u0430\u0434\u0443\u0441\u043e\u0432).<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\"><span style=\"color: #C678DD\">import<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">numpy<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #C678DD\">as<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">np<\/span>\n\n<span style=\"color: #E06C75\">a<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">array<\/span><span style=\"color: #ABB2BF\">([<\/span><span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">0<\/span><span style=\"color: #ABB2BF\">])<\/span>\n<span style=\"color: #E06C75\">b<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">array<\/span><span style=\"color: #ABB2BF\">([<\/span><span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">])<\/span>\n\n<span style=\"color: #E06C75\">cos<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">dot<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">a<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">b<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #56B6C2\">\/<\/span> <span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">linalg<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">norm<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">a<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #56B6C2\">*<\/span> <span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">linalg<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">norm<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">b<\/span><span style=\"color: #ABB2BF\">))<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">cos<\/span><span style=\"color: #ABB2BF\">)<\/span>                          <span style=\"color: #7F848E\"># 0.7071067811865475<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">degrees<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">arccos<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">cos<\/span><span style=\"color: #ABB2BF\">)))<\/span>   <span style=\"color: #7F848E\"># 45.00000000000001<\/span><\/code><\/pre>\n<p>\u041a\u043e\u0441\u0438\u043d\u0443\u0441 \u0432\u044b\u0448\u0435\u043b <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">0.7071...<\/code> (\u044d\u0442\u043e 1\/sqrt(2)), \u0430 \u0443\u0433\u043e\u043b &#8212; \u043f\u0440\u0438\u043c\u0435\u0440\u043d\u043e <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">45<\/code> \u0433\u0440\u0430\u0434\u0443\u0441\u043e\u0432. \u0425\u0432\u043e\u0441\u0442 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">...0001<\/code> \u0432 \u0432\u044b\u0432\u043e\u0434\u0435 \u0443\u0433\u043b\u0430 &#8212; \u043e\u0431\u044b\u0447\u043d\u0430\u044f \u043f\u043e\u0433\u0440\u0435\u0448\u043d\u043e\u0441\u0442\u044c \u0447\u0438\u0441\u0435\u043b \u0441 \u043f\u043b\u0430\u0432\u0430\u044e\u0449\u0435\u0439 \u0442\u043e\u0447\u043a\u043e\u0439, \u0430 \u043d\u0435 \u043e\u0448\u0438\u0431\u043a\u0430: \u0442\u043e\u0447\u043d\u043e\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \u0437\u0434\u0435\u0441\u044c 45 \u0433\u0440\u0430\u0434\u0443\u0441\u043e\u0432. \u0418\u043c\u0435\u043d\u043d\u043e \u044d\u0442\u0430 \u0444\u043e\u0440\u043c\u0443\u043b\u0430 \u043b\u0435\u0436\u0438\u0442 \u0432 \u043e\u0441\u043d\u043e\u0432\u0435 \u043a\u043e\u0441\u0438\u043d\u0443\u0441\u043d\u043e\u0439 \u0431\u043b\u0438\u0437\u043e\u0441\u0442\u0438 \u0432 \u043c\u0430\u0448\u0438\u043d\u043d\u043e\u043c \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0438 &#8212; \u043e \u043d\u0435\u0439 \u043d\u0438\u0436\u0435.<\/p>\n<h2 id=\"s6\">\u041d\u043e\u0440\u043c\u0430\u043b\u0438\u0437\u0430\u0446\u0438\u044f \u0432\u0435\u043a\u0442\u043e\u0440\u0430<\/h2>\n<p><strong>\u041d\u043e\u0440\u043c\u0430\u043b\u0438\u0437\u0430\u0446\u0438\u044f<\/strong> &#8212; \u044d\u0442\u043e \u043f\u0440\u0438\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0432\u0435\u043a\u0442\u043e\u0440\u0430 \u043a \u0435\u0434\u0438\u043d\u0438\u0447\u043d\u043e\u0439 \u0434\u043b\u0438\u043d\u0435 \u0441 \u0441\u043e\u0445\u0440\u0430\u043d\u0435\u043d\u0438\u0435\u043c \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u044f. \u0415\u0434\u0438\u043d\u0438\u0447\u043d\u044b\u0439 \u0432\u0435\u043a\u0442\u043e\u0440 \u043f\u043e\u043b\u0443\u0447\u0430\u044e\u0442 \u0434\u0435\u043b\u0435\u043d\u0438\u0435\u043c \u0438\u0441\u0445\u043e\u0434\u043d\u043e\u0433\u043e \u043d\u0430 \u0435\u0433\u043e \u0434\u043b\u0438\u043d\u0443:<\/p>\n<p>u = a \/ |a|<\/p>\n<p>\u041e\u043f\u0435\u0440\u0430\u0446\u0438\u044f \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0430 \u0442\u043e\u043b\u044c\u043a\u043e \u0434\u043b\u044f \u043d\u0435\u043d\u0443\u043b\u0435\u0432\u043e\u0433\u043e \u0432\u0435\u043a\u0442\u043e\u0440\u0430: \u0443 \u043d\u0443\u043b\u0435\u0432\u043e\u0433\u043e \u0434\u043b\u0438\u043d\u0430 \u0440\u0430\u0432\u043d\u0430 \u043d\u0443\u043b\u044e, \u0438 \u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043d\u0435\u0432\u043e\u0437\u043c\u043e\u0436\u043d\u043e. \u0415\u0434\u0438\u043d\u0438\u0447\u043d\u044b\u0435 \u0432\u0435\u043a\u0442\u043e\u0440\u044b \u0443\u0434\u043e\u0431\u043d\u044b \u0442\u0430\u043c, \u0433\u0434\u0435 \u0432\u0430\u0436\u043d\u043e \u0442\u043e\u043b\u044c\u043a\u043e \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435, \u0430 \u043d\u0435 \u043c\u0430\u0441\u0448\u0442\u0430\u0431 (\u043d\u0430\u043f\u0440\u0438\u043c\u0435\u0440 \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043e\u0441\u0432\u0435\u0449\u0435\u043d\u0438\u044f \u0432 \u0433\u0440\u0430\u0444\u0438\u043a\u0435).<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\"><span style=\"color: #C678DD\">import<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">numpy<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #C678DD\">as<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">np<\/span>\n\n<span style=\"color: #E06C75\">v<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">array<\/span><span style=\"color: #ABB2BF\">([<\/span><span style=\"color: #D19A66\">3.0<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">4.0<\/span><span style=\"color: #ABB2BF\">])<\/span>\n<span style=\"color: #E06C75\">unit<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">v<\/span> <span style=\"color: #56B6C2\">\/<\/span> <span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">linalg<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">norm<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">v<\/span><span style=\"color: #ABB2BF\">)<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">unit<\/span><span style=\"color: #ABB2BF\">)<\/span>                   <span style=\"color: #7F848E\"># [0.6 0.8]<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">linalg<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">norm<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">unit<\/span><span style=\"color: #ABB2BF\">))<\/span>   <span style=\"color: #7F848E\"># 1.0<\/span><\/code><\/pre>\n<p>\u0418\u0437 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">[3, 4]<\/code> \u043f\u043e\u043b\u0443\u0447\u0438\u043b\u0441\u044f <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">[0.6, 0.8]<\/code>, \u0438 \u0435\u0433\u043e \u0434\u043b\u0438\u043d\u0430 \u0440\u0430\u0432\u043d\u0430 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">1.0<\/code> (\u0441 \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u044c\u044e \u0434\u043e \u043f\u043e\u0433\u0440\u0435\u0448\u043d\u043e\u0441\u0442\u0438 float). \u041d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043e\u0441\u0442\u0430\u043b\u043e\u0441\u044c \u043f\u0440\u0435\u0436\u043d\u0438\u043c, \u0438\u0437\u043c\u0435\u043d\u0438\u043b\u0441\u044f \u0442\u043e\u043b\u044c\u043a\u043e \u043c\u0430\u0441\u0448\u0442\u0430\u0431. \u0412\u0430\u0436\u043d\u043e \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c \u043e\u0431\u044b\u0447\u043d\u043e\u0435 \u0434\u0435\u043b\u0435\u043d\u0438\u0435 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">\/<\/code> (\u043e\u043d\u043e \u0432\u043e\u0437\u0432\u0440\u0430\u0449\u0430\u0435\u0442 float): \u0446\u0435\u043b\u043e\u0447\u0438\u0441\u043b\u0435\u043d\u043d\u043e\u0435 \u0434\u0435\u043b\u0435\u043d\u0438\u0435 <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">\/\/<\/code> \u0432 \u0440\u0443\u0447\u043d\u043e\u043c \u043a\u043e\u0434\u0435 \u043e\u0431\u043d\u0443\u043b\u0438\u043b\u043e \u0431\u044b \u0434\u0440\u043e\u0431\u043d\u0443\u044e \u0447\u0430\u0441\u0442\u044c.<\/p>\n<h2 id=\"s7\">\u0412\u0435\u043a\u0442\u043e\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 (\u0442\u043e\u043b\u044c\u043a\u043e \u0432 3D)<\/h2>\n<p><strong>\u0412\u0435\u043a\u0442\u043e\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435<\/strong> (cross product) \u0434\u0432\u0443\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432 \u0434\u0430\u0435\u0442 \u043d\u0435 \u0447\u0438\u0441\u043b\u043e, \u0430 \u043d\u043e\u0432\u044b\u0439 \u0432\u0435\u043a\u0442\u043e\u0440, \u043f\u0435\u0440\u043f\u0435\u043d\u0434\u0438\u043a\u0443\u043b\u044f\u0440\u043d\u044b\u0439 \u043e\u0431\u043e\u0438\u043c \u0438\u0441\u0445\u043e\u0434\u043d\u044b\u043c. \u0412\u0430\u0436\u043d\u0430\u044f \u0433\u0440\u0430\u043d\u0438\u0446\u0430: \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044f \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0430 \u0438\u043c\u0435\u043d\u043d\u043e \u0434\u043b\u044f \u0442\u0440\u0435\u0445\u043c\u0435\u0440\u043d\u044b\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432 (\u0441\u0442\u0440\u043e\u0433\u043e \u0433\u043e\u0432\u043e\u0440\u044f, \u0435\u0449\u0435 \u0438 \u0432 \u0441\u0435\u043c\u0438\u043c\u0435\u0440\u043d\u043e\u043c \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0435, \u043d\u043e \u043d\u0430 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u0435 \u044d\u0442\u043e 3D). \u0414\u043b\u044f \u0434\u0432\u0443\u043c\u0435\u0440\u043d\u044b\u0445 \u0438\u043b\u0438 \u043c\u043d\u043e\u0433\u043e\u043c\u0435\u0440\u043d\u044b\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432 \u00ab\u0432\u0435\u043a\u0442\u043e\u0440\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f\u00bb \u0432 \u043e\u0431\u044b\u0447\u043d\u043e\u043c \u0441\u043c\u044b\u0441\u043b\u0435 \u043d\u0435\u0442.<\/p>\n<p>\u0414\u043b\u0438\u043d\u0430 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u0430 \u0441\u0432\u044f\u0437\u0430\u043d\u0430 \u0441 \u0441\u0438\u043d\u0443\u0441\u043e\u043c \u0443\u0433\u043b\u0430: <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">|a x b| = |a| * |b| * sin(theta)<\/code>. \u0415\u0441\u043b\u0438 \u0432\u0435\u043a\u0442\u043e\u0440\u044b \u043f\u0430\u0440\u0430\u043b\u043b\u0435\u043b\u044c\u043d\u044b (\u0443\u0433\u043e\u043b 0 \u0438\u043b\u0438 180 \u0433\u0440\u0430\u0434\u0443\u0441\u043e\u0432), \u0438\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 &#8212; \u043d\u0443\u043b\u0435\u0432\u043e\u0439 \u0432\u0435\u043a\u0442\u043e\u0440.<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\"><span style=\"color: #C678DD\">import<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">numpy<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #C678DD\">as<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">np<\/span>\n\n<span style=\"color: #E06C75\">a<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">array<\/span><span style=\"color: #ABB2BF\">([<\/span><span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">0<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">0<\/span><span style=\"color: #ABB2BF\">])<\/span>\n<span style=\"color: #E06C75\">b<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">array<\/span><span style=\"color: #ABB2BF\">([<\/span><span style=\"color: #D19A66\">0<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">0<\/span><span style=\"color: #ABB2BF\">])<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">cross<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">a<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">b<\/span><span style=\"color: #ABB2BF\">))<\/span>  <span style=\"color: #7F848E\"># [0 0 1]<\/span><\/code><\/pre>\n<p>\u0412\u0435\u043a\u0442\u043e\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u043e\u0440\u0442\u0430 \u043e\u0441\u0438 X \u043d\u0430 \u043e\u0440\u0442 \u043e\u0441\u0438 Y \u0434\u0430\u043b\u043e <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">[0, 0, 1]<\/code> &#8212; \u043e\u0440\u0442 \u043e\u0441\u0438 Z, \u0442\u043e \u0435\u0441\u0442\u044c \u0432\u0435\u043a\u0442\u043e\u0440, \u043f\u0435\u0440\u043f\u0435\u043d\u0434\u0438\u043a\u0443\u043b\u044f\u0440\u043d\u044b\u0439 \u043f\u043b\u043e\u0441\u043a\u043e\u0441\u0442\u0438 XY. \u042d\u0442\u043e \u043a\u043b\u044e\u0447\u0435\u0432\u0430\u044f \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044f \u0432 3D-\u0433\u0440\u0430\u0444\u0438\u043a\u0435 \u0438 \u0444\u0438\u0437\u0438\u043a\u0435: \u0442\u0430\u043a \u0432\u044b\u0447\u0438\u0441\u043b\u044f\u044e\u0442 \u043d\u043e\u0440\u043c\u0430\u043b\u044c \u043a \u043f\u043e\u0432\u0435\u0440\u0445\u043d\u043e\u0441\u0442\u0438 \u0438 \u043c\u043e\u043c\u0435\u043d\u0442 \u0441\u0438\u043b\u044b.<\/p>\n<p>\u041d\u0435 \u043f\u0443\u0442\u0430\u0439\u0442\u0435 \u0434\u0432\u0435 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0438 \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u044f: \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 (<code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">np.dot<\/code>) \u0434\u0430\u0435\u0442 \u0447\u0438\u0441\u043b\u043e \u0438 \u0440\u0430\u0431\u043e\u0442\u0430\u0435\u0442 \u0432 \u043b\u044e\u0431\u043e\u0439 \u0440\u0430\u0437\u043c\u0435\u0440\u043d\u043e\u0441\u0442\u0438; \u0432\u0435\u043a\u0442\u043e\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 (<code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">np.cross<\/code>) \u0434\u0430\u0435\u0442 \u0432\u0435\u043a\u0442\u043e\u0440 \u0438 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043e \u0442\u043e\u043b\u044c\u043a\u043e \u0432 3D.<\/p>\n<table>\n<thead>\n<tr>\n<th>\u041e\u043f\u0435\u0440\u0430\u0446\u0438\u044f<\/th>\n<th>\u0412\u0445\u043e\u0434<\/th>\n<th>\u0420\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442<\/th>\n<th>\u0420\u0430\u0437\u043c\u0435\u0440\u043d\u043e\u0441\u0442\u044c<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\u0421\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 (dot)<\/td>\n<td>\u0434\u0432\u0430 \u0432\u0435\u043a\u0442\u043e\u0440\u0430<\/td>\n<td>\u0447\u0438\u0441\u043b\u043e (\u0441\u043a\u0430\u043b\u044f\u0440)<\/td>\n<td>\u043b\u044e\u0431\u0430\u044f<\/td>\n<\/tr>\n<tr>\n<td>\u0412\u0435\u043a\u0442\u043e\u0440\u043d\u043e\u0435 (cross)<\/td>\n<td>\u0434\u0432\u0430 \u0432\u0435\u043a\u0442\u043e\u0440\u0430<\/td>\n<td>\u0432\u0435\u043a\u0442\u043e\u0440, \u043f\u0435\u0440\u043f\u0435\u043d\u0434\u0438\u043a\u0443\u043b\u044f\u0440\u043d\u044b\u0439 \u043e\u0431\u043e\u0438\u043c<\/td>\n<td>\u0442\u043e\u043b\u044c\u043a\u043e 3D<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 id=\"s8\">\u0413\u0434\u0435 \u0432\u0435\u043a\u0442\u043e\u0440\u044b \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u044e\u0442\u0441\u044f<\/h2>\n<p><strong>\u041c\u0430\u0448\u0438\u043d\u043d\u043e\u0435 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435.<\/strong> \u0417\u0434\u0435\u0441\u044c \u0432\u0435\u043a\u0442\u043e\u0440 &#8212; \u044d\u0442\u043e \u0443\u043f\u043e\u0440\u044f\u0434\u043e\u0447\u0435\u043d\u043d\u044b\u0439 \u043d\u0430\u0431\u043e\u0440 \u0447\u0438\u0441\u0435\u043b, \u043e\u043f\u0438\u0441\u044b\u0432\u0430\u044e\u0449\u0438\u0439 \u043e\u0431\u044a\u0435\u043a\u0442. \u041f\u0440\u0438\u0437\u043d\u0430\u043a\u0438 \u043e\u0431\u044a\u0435\u043a\u0442\u0430 (\u0440\u043e\u0441\u0442, \u0432\u0435\u0441, \u0432\u043e\u0437\u0440\u0430\u0441\u0442) \u0441\u043a\u043b\u0430\u0434\u044b\u0432\u0430\u044e\u0442 \u0432 \u0432\u0435\u043a\u0442\u043e\u0440 \u043f\u0440\u0438\u0437\u043d\u0430\u043a\u043e\u0432. \u0422\u0435\u043a\u0441\u0442, \u043a\u0430\u0440\u0442\u0438\u043d\u043a\u0443 \u0438\u043b\u0438 \u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u0435\u043b\u044f \u043f\u0435\u0440\u0435\u0432\u043e\u0434\u044f\u0442 \u0432 \u044d\u043c\u0431\u0435\u0434\u0434\u0438\u043d\u0433 &#8212; \u043f\u043b\u043e\u0442\u043d\u044b\u0439 \u0447\u0438\u0441\u043b\u043e\u0432\u043e\u0439 \u0432\u0435\u043a\u0442\u043e\u0440 \u0431\u043e\u043b\u044c\u0448\u043e\u0439 \u0440\u0430\u0437\u043c\u0435\u0440\u043d\u043e\u0441\u0442\u0438 (\u0441\u043e\u0442\u043d\u0438 \u0438 \u0442\u044b\u0441\u044f\u0447\u0438 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442). \u0411\u043b\u0438\u0437\u043e\u0441\u0442\u044c \u043e\u0431\u044a\u0435\u043a\u0442\u043e\u0432 \u0438\u0437\u043c\u0435\u0440\u044f\u044e\u0442 \u043a\u043e\u0441\u0438\u043d\u0443\u0441\u043d\u043e\u0439 \u0431\u043b\u0438\u0437\u043e\u0441\u0442\u044c\u044e &#8212; \u043a\u043e\u0441\u0438\u043d\u0443\u0441\u043e\u043c \u0443\u0433\u043b\u0430 \u043c\u0435\u0436\u0434\u0443 \u0438\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438: \u0447\u0435\u043c \u0431\u043b\u0438\u0436\u0435 \u043a 1, \u0442\u0435\u043c \u0431\u043e\u043b\u0435\u0435 \u043f\u043e\u0445\u043e\u0436\u0438 \u043e\u0431\u044a\u0435\u043a\u0442\u044b. \u041d\u0430 \u044d\u0442\u043e\u043c \u0440\u0430\u0431\u043e\u0442\u0430\u044e\u0442 \u043f\u043e\u0438\u0441\u043a \u043f\u043e\u0445\u043e\u0436\u0438\u0445 \u0442\u043e\u0432\u0430\u0440\u043e\u0432 \u0438 \u0440\u0435\u043a\u043e\u043c\u0435\u043d\u0434\u0430\u0446\u0438\u0438.<\/p>\n<pre style=\"background:#282c34;color:#abb2bf;border-radius:8px;padding:16px 20px;margin:20px 0;overflow-x:auto;font-size:14px;line-height:1.55;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\"><code style=\"background:none\"><span style=\"color: #C678DD\">import<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">numpy<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #C678DD\">as<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #E06C75\">np<\/span>\n\n<span style=\"color: #C678DD\">def<\/span><span style=\"color: #ABB2BF\"> <\/span><span style=\"color: #61AFEF; font-weight: bold\">cosine_sim<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">a<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">b<\/span><span style=\"color: #ABB2BF\">):<\/span>\n    <span style=\"color: #C678DD\">return<\/span> <span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">dot<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">a<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">b<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #56B6C2\">\/<\/span> <span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">linalg<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">norm<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">a<\/span><span style=\"color: #ABB2BF\">)<\/span> <span style=\"color: #56B6C2\">*<\/span> <span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">linalg<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">norm<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">b<\/span><span style=\"color: #ABB2BF\">))<\/span>\n\n<span style=\"color: #E06C75\">u1<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">array<\/span><span style=\"color: #ABB2BF\">([<\/span><span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">0<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">])<\/span>   <span style=\"color: #7F848E\"># \u044d\u043c\u0431\u0435\u0434\u0434\u0438\u043d\u0433 \u043e\u0431\u044a\u0435\u043a\u0442\u0430 1<\/span>\n<span style=\"color: #E06C75\">u2<\/span> <span style=\"color: #56B6C2\">=<\/span> <span style=\"color: #E06C75\">np<\/span><span style=\"color: #56B6C2\">.<\/span><span style=\"color: #E06C75\">array<\/span><span style=\"color: #ABB2BF\">([<\/span><span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #D19A66\">1<\/span><span style=\"color: #ABB2BF\">])<\/span>   <span style=\"color: #7F848E\"># \u044d\u043c\u0431\u0435\u0434\u0434\u0438\u043d\u0433 \u043e\u0431\u044a\u0435\u043a\u0442\u0430 2<\/span>\n<span style=\"color: #E5C07B\">print<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E5C07B\">round<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">cosine_sim<\/span><span style=\"color: #ABB2BF\">(<\/span><span style=\"color: #E06C75\">u1<\/span><span style=\"color: #ABB2BF\">,<\/span> <span style=\"color: #E06C75\">u2<\/span><span style=\"color: #ABB2BF\">),<\/span> <span style=\"color: #D19A66\">3<\/span><span style=\"color: #ABB2BF\">))<\/span>  <span style=\"color: #7F848E\"># 0.816<\/span><\/code><\/pre>\n<p>\u041a\u043e\u0441\u0438\u043d\u0443\u0441\u043d\u0430\u044f \u0431\u043b\u0438\u0437\u043e\u0441\u0442\u044c \u0432\u044b\u0448\u043b\u0430 \u043f\u0440\u0438\u043c\u0435\u0440\u043d\u043e <code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">0.816<\/code> &#8212; \u043e\u0431\u044a\u0435\u043a\u0442\u044b \u043f\u043e\u0445\u043e\u0436\u0438 \u043f\u043e \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u044e. \u0424\u0443\u043d\u043a\u0446\u0438\u044f \u043d\u0435 \u0437\u043d\u0430\u0435\u0442 \u0441\u043c\u044b\u0441\u043b\u0430 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442, \u043e\u043d\u0430 \u0441\u0440\u0430\u0432\u043d\u0438\u0432\u0430\u0435\u0442 \u0442\u043e\u043b\u044c\u043a\u043e \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044e \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432. \u0414\u043b\u044f \u0440\u0435\u0430\u043b\u044c\u043d\u044b\u0445 \u044d\u043c\u0431\u0435\u0434\u0434\u0438\u043d\u0433\u043e\u0432 \u0440\u0430\u0437\u043c\u0435\u0440\u043d\u043e\u0441\u0442\u044c \u0431\u0443\u0434\u0435\u0442 \u043d\u0435 3, \u0430 \u0441\u043e\u0442\u043d\u0438, \u043d\u043e \u0444\u043e\u0440\u043c\u0443\u043b\u0430 \u0442\u0430 \u0436\u0435.<\/p>\n<p><strong>\u041a\u043e\u043c\u043f\u044c\u044e\u0442\u0435\u0440\u043d\u0430\u044f \u0433\u0440\u0430\u0444\u0438\u043a\u0430.<\/strong> \u0412\u0435\u043a\u0442\u043e\u0440\u044b \u0437\u0430\u0434\u0430\u044e\u0442 \u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0432\u0435\u0440\u0448\u0438\u043d, \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u044f \u0438 \u043f\u0435\u0440\u0435\u043c\u0435\u0449\u0435\u043d\u0438\u044f. \u0412 \u0432\u0435\u043a\u0442\u043e\u0440\u043d\u043e\u0439 \u0433\u0440\u0430\u0444\u0438\u043a\u0435 (\u0438\u043a\u043e\u043d\u043a\u0438, \u043b\u043e\u0433\u043e\u0442\u0438\u043f\u044b, \u0448\u0440\u0438\u0444\u0442\u044b) \u0438\u0437\u043e\u0431\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u043e\u043f\u0438\u0441\u0430\u043d\u043e \u043d\u0435 \u043f\u0438\u043a\u0441\u0435\u043b\u044f\u043c\u0438, \u0430 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0435\u0439 &#8212; \u043b\u0438\u043d\u0438\u044f\u043c\u0438 \u0438 \u043a\u0440\u0438\u0432\u044b\u043c\u0438 \u043f\u043e \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u0430\u043c, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0435\u0433\u043e \u043c\u043e\u0436\u043d\u043e \u0443\u0432\u0435\u043b\u0438\u0447\u0438\u0432\u0430\u0442\u044c \u0431\u0435\u0437 \u043f\u043e\u0442\u0435\u0440\u0438 \u0447\u0435\u0442\u043a\u043e\u0441\u0442\u0438. \u0412 3D \u043d\u043e\u0440\u043c\u0430\u043b\u044c \u043a \u043f\u043e\u0432\u0435\u0440\u0445\u043d\u043e\u0441\u0442\u0438 (\u0447\u0435\u0440\u0435\u0437 \u0432\u0435\u043a\u0442\u043e\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435) \u0438 \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u0441\u0432\u0435\u0442\u0430 (\u0435\u0434\u0438\u043d\u0438\u0447\u043d\u044b\u0439 \u0432\u0435\u043a\u0442\u043e\u0440 \u043f\u043e\u0441\u043b\u0435 \u043d\u043e\u0440\u043c\u0430\u043b\u0438\u0437\u0430\u0446\u0438\u0438) \u043d\u0443\u0436\u043d\u044b, \u0447\u0442\u043e\u0431\u044b \u0440\u0430\u0441\u0441\u0447\u0438\u0442\u0430\u0442\u044c \u043e\u0441\u0432\u0435\u0449\u0435\u043d\u0438\u0435.<\/p>\n<p><strong>\u0424\u0438\u0437\u0438\u043a\u0430 \u0438 \u0438\u043d\u0436\u0435\u043d\u0435\u0440\u0438\u044f.<\/strong> \u0421\u043a\u043e\u0440\u043e\u0441\u0442\u044c, \u0441\u0438\u043b\u0430, \u0443\u0441\u043a\u043e\u0440\u0435\u043d\u0438\u0435 &#8212; \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u044b \u0441 \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435\u043c, \u0438\u0445 \u0435\u0441\u0442\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u043e \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u0442\u044c \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438: \u0441\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432 \u0434\u0430\u0435\u0442 \u0441\u0443\u043c\u043c\u0430\u0440\u043d\u0443\u044e \u0441\u0438\u043b\u0443 \u0438\u043b\u0438 \u0441\u043a\u043e\u0440\u043e\u0441\u0442\u044c, \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0441\u0438\u043b\u044b \u043d\u0430 \u043f\u0435\u0440\u0435\u043c\u0435\u0449\u0435\u043d\u0438\u0435 &#8212; \u0440\u0430\u0431\u043e\u0442\u0443, \u0430 \u0432\u0435\u043a\u0442\u043e\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 &#8212; \u043c\u043e\u043c\u0435\u043d\u0442 \u0441\u0438\u043b\u044b.<\/p>\n<h2 id=\"s9\">\u0412\u044b\u0432\u043e\u0434\u044b<\/h2>\n<ul>\n<li>\u0412\u0435\u043a\u0442\u043e\u0440 \u0438\u043c\u0435\u0435\u0442 \u0434\u0432\u0430 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u0438\u044f: \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u043e\u0435 (\u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u043d\u044b\u0439 \u043e\u0442\u0440\u0435\u0437\u043e\u043a) \u0438 \u0430\u043b\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043a\u043e\u0435 (\u0443\u043f\u043e\u0440\u044f\u0434\u043e\u0447\u0435\u043d\u043d\u044b\u0439 \u043d\u0430\u0431\u043e\u0440 \u0447\u0438\u0441\u0435\u043b); \u0432 \u043c\u0430\u0448\u0438\u043d\u043d\u043e\u043c \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0438 \u043f\u043e\u0447\u0442\u0438 \u0432\u0441\u0435\u0433\u0434\u0430 \u0440\u0430\u0431\u043e\u0442\u0430\u044e\u0442 \u0441\u043e \u0432\u0442\u043e\u0440\u044b\u043c.<\/li>\n<li>\u0411\u0430\u0437\u043e\u0432\u044b\u0435 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0438 &#8212; \u0441\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0438 \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u0441\u043a\u0430\u043b\u044f\u0440 &#8212; \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u044e\u0442\u0441\u044f \u043f\u043e\u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u043d\u043e; \u0434\u043b\u0438\u043d\u0430 (\u043d\u043e\u0440\u043c\u0430) \u0441\u0447\u0438\u0442\u0430\u0435\u0442\u0441\u044f \u043a\u0430\u043a \u043a\u043e\u0440\u0435\u043d\u044c \u0438\u0437 \u0441\u0443\u043c\u043c\u044b \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043e\u0432 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442.<\/li>\n<li>\u0421\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0434\u0430\u0435\u0442 \u0447\u0438\u0441\u043b\u043e \u0438 \u0440\u0430\u0431\u043e\u0442\u0430\u0435\u0442 \u0432 \u043b\u044e\u0431\u043e\u0439 \u0440\u0430\u0437\u043c\u0435\u0440\u043d\u043e\u0441\u0442\u0438; \u0432\u0435\u043a\u0442\u043e\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0434\u0430\u0435\u0442 \u043f\u0435\u0440\u043f\u0435\u043d\u0434\u0438\u043a\u0443\u043b\u044f\u0440\u043d\u044b\u0439 \u0432\u0435\u043a\u0442\u043e\u0440 \u0438 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043e \u0442\u043e\u043b\u044c\u043a\u043e \u0432 3D &#8212; \u044d\u0442\u043e \u0440\u0430\u0437\u043d\u044b\u0435 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u0438.<\/li>\n<li>\u0423\u0433\u043e\u043b \u043c\u0435\u0436\u0434\u0443 \u0432\u0435\u043a\u0442\u043e\u0440\u0430\u043c\u0438 \u0438 \u043a\u043e\u0441\u0438\u043d\u0443\u0441\u043d\u0430\u044f \u0431\u043b\u0438\u0437\u043e\u0441\u0442\u044c \u0441\u0447\u0438\u0442\u0430\u044e\u0442\u0441\u044f \u0447\u0435\u0440\u0435\u0437 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0438 \u043d\u043e\u0440\u043c\u044b; \u0444\u043e\u0440\u043c\u0443\u043b\u0430 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0430 \u0442\u043e\u043b\u044c\u043a\u043e \u0434\u043b\u044f \u043d\u0435\u043d\u0443\u043b\u0435\u0432\u044b\u0445 \u0432\u0435\u043a\u0442\u043e\u0440\u043e\u0432.<\/li>\n<li>\u0412\u0435\u043a\u0442\u043e\u0440\u044b &#8212; \u043e\u0441\u043d\u043e\u0432\u0430 \u043f\u0440\u0438\u0437\u043d\u0430\u043a\u043e\u0432 \u0438 \u044d\u043c\u0431\u0435\u0434\u0434\u0438\u043d\u0433\u043e\u0432 \u0432 ML, \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0438 \u0432 \u043a\u043e\u043c\u043f\u044c\u044e\u0442\u0435\u0440\u043d\u043e\u0439 \u0433\u0440\u0430\u0444\u0438\u043a\u0435 \u0438 \u043e\u043f\u0438\u0441\u0430\u043d\u0438\u044f \u0441\u0438\u043b \u0438 \u0441\u043a\u043e\u0440\u043e\u0441\u0442\u0435\u0439 \u0432 \u0444\u0438\u0437\u0438\u043a\u0435.<\/li>\n<\/ul>\n<h2 id=\"s10\">\u0413\u0434\u0435 \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u0435\u0442\u0441\u044f \/ \u0441\u0432\u044f\u0437\u044c \u0441 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u043e\u0439<\/h2>\n<div class=\"oj-cta\" style=\"background:#fbf4e4;border-left:4px solid #c9962b;border-radius:8px;padding:16px 20px;margin:24px 0\">\n<p style=\"margin:0 0 8px\"><strong>\u041e\u0441\u0432\u043e\u0439\u0442\u0435 \u0442\u0435\u043c\u0443 \u043d\u0430 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u0435<\/strong><\/p>\n<p>\u0412\u0435\u043a\u0442\u043e\u0440\u044b, \u043d\u043e\u0440\u043c\u044b \u0438 \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 &#8212; \u044d\u0442\u043e \u0444\u0443\u043d\u0434\u0430\u043c\u0435\u043d\u0442 \u043c\u0430\u0448\u0438\u043d\u043d\u043e\u0433\u043e \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u044f: \u043f\u0440\u0438\u0437\u043d\u0430\u043a\u0438, \u044d\u043c\u0431\u0435\u0434\u0434\u0438\u043d\u0433\u0438, \u043c\u0435\u0442\u0440\u0438\u043a\u0438 \u0431\u043b\u0438\u0437\u043e\u0441\u0442\u0438, \u0430 \u0434\u0430\u043b\u044c\u0448\u0435 \u043c\u0430\u0442\u0440\u0438\u0446\u044b \u0438 \u0433\u0440\u0430\u0434\u0438\u0435\u043d\u0442\u044b. \u0411\u0435\u0437 \u0443\u0432\u0435\u0440\u0435\u043d\u043d\u043e\u0433\u043e \u0432\u043b\u0430\u0434\u0435\u043d\u0438\u044f \u044d\u0442\u0438\u043c\u0438 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044f\u043c\u0438 \u0441\u043b\u043e\u0436\u043d\u043e \u043f\u043e\u043d\u044f\u0442\u044c, \u043a\u0430\u043a \u043c\u043e\u0434\u0435\u043b\u044c \u0445\u0440\u0430\u043d\u0438\u0442 \u0434\u0430\u043d\u043d\u044b\u0435 \u0438 \u0441\u0447\u0438\u0442\u0430\u0435\u0442 \u0440\u0430\u0441\u0441\u0442\u043e\u044f\u043d\u0438\u044f \u043c\u0435\u0436\u0434\u0443 \u043e\u0431\u044a\u0435\u043a\u0442\u0430\u043c\u0438. \u0427\u0442\u043e\u0431\u044b \u0441\u0438\u0441\u0442\u0435\u043c\u043d\u043e \u0434\u043e\u0439\u0442\u0438 \u043e\u0442 \u0431\u0430\u0437\u043e\u0432\u043e\u0439 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0438 \u0434\u043e \u0440\u0430\u0431\u043e\u0447\u0438\u0445 ML-\u043c\u043e\u0434\u0435\u043b\u0435\u0439, \u043f\u043e\u0441\u043c\u043e\u0442\u0440\u0438\u0442\u0435 \u043a\u0443\u0440\u0441 <a href=\"https:\/\/otus.ru\/lessons\/ml-basic\/?int_article=vse-o-vektorah-ot-a-do-ya&amp;int_place=article&amp;int_variant=v1\">ML-\u0441\u0442\u0430\u0440\u0442: \u0431\u0430\u0437\u043e\u0432\u044b\u0439 \u043a\u0443\u0440\u0441 \u043f\u043e \u043c\u0430\u0448\u0438\u043d\u043d\u043e\u043c\u0443 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u044e<\/a>, \u0433\u0434\u0435 \u043b\u0438\u043d\u0435\u0439\u043d\u0430\u044f \u0430\u043b\u0433\u0435\u0431\u0440\u0430 \u0440\u0430\u0437\u0431\u0438\u0440\u0430\u0435\u0442\u0441\u044f \u043d\u0430 \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0437\u0430\u0434\u0430\u0447\u0430\u0445.<\/p>\n<\/div>\n<div class=\"oj-cta\" style=\"background:#fbf4e4;border-left:4px solid #c9962b;border-radius:8px;padding:16px 20px;margin:24px 0\">\n<p style=\"margin:0 0 8px\"><strong>\u041e\u0441\u0432\u043e\u0439\u0442\u0435 \u0442\u0435\u043c\u0443 \u043d\u0430 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u0435<\/strong><\/p>\n<p>\u041e\u0446\u0435\u043d\u0438\u0442\u044c \u0444\u043e\u0440\u043c\u0430\u0442 \u0438 \u0443\u0440\u043e\u0432\u0435\u043d\u044c \u043f\u043e\u0434\u0430\u0447\u0438 \u043f\u043e\u043c\u043e\u0433\u0443\u0442 \u0431\u0435\u0441\u043f\u043b\u0430\u0442\u043d\u044b\u0435 <a href=\"https:\/\/otus.ru\/events\/?int_article=vse-o-vektorah-ot-a-do-ya&amp;int_place=article&amp;int_variant=v1\">\u043e\u0442\u043a\u0440\u044b\u0442\u044b\u0435 \u0443\u0440\u043e\u043a\u0438 Otus<\/a> &#8212; \u043d\u0430 \u043d\u0438\u0445 \u043c\u043e\u0436\u043d\u043e \u0432\u0436\u0438\u0432\u0443\u044e \u0437\u0430\u0434\u0430\u0442\u044c \u0432\u043e\u043f\u0440\u043e\u0441 \u043f\u0440\u0435\u043f\u043e\u0434\u0430\u0432\u0430\u0442\u0435\u043b\u044e.<\/p>\n<\/div>\n<h2 id=\"s11\">FAQ<\/h2>\n<p><strong>\u0427\u0435\u043c \u0432\u0435\u043a\u0442\u043e\u0440 \u043e\u0442\u043b\u0438\u0447\u0430\u0435\u0442\u0441\u044f \u043e\u0442 \u043c\u0430\u0441\u0441\u0438\u0432\u0430 \u0432 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u043c\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u0438?<\/strong><br \/>\n\u041c\u0430\u0441\u0441\u0438\u0432 &#8212; \u044d\u0442\u043e \u0441\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u0430 \u0434\u0430\u043d\u043d\u044b\u0445 (\u043a\u043e\u043d\u0442\u0435\u0439\u043d\u0435\u0440) \u0434\u043b\u044f \u0445\u0440\u0430\u043d\u0435\u043d\u0438\u044f \u044d\u043b\u0435\u043c\u0435\u043d\u0442\u043e\u0432, \u0430 \u0432\u0435\u043a\u0442\u043e\u0440 &#8212; \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u043e\u0431\u044a\u0435\u043a\u0442 \u0441 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043d\u044b\u043c\u0438 \u043e\u043f\u0435\u0440\u0430\u0446\u0438\u044f\u043c\u0438 (\u0441\u043b\u043e\u0436\u0435\u043d\u0438\u0435, \u043d\u043e\u0440\u043c\u0430, \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f). \u041a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u044b \u0432\u0435\u043a\u0442\u043e\u0440\u0430 \u0443\u0434\u043e\u0431\u043d\u043e \u0445\u0440\u0430\u043d\u0438\u0442\u044c \u0432 \u043c\u0430\u0441\u0441\u0438\u0432\u0435, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \u0432 \u043a\u043e\u0434\u0435 \u0432\u0435\u043a\u0442\u043e\u0440 \u0447\u0430\u0441\u0442\u043e \u0438 \u0435\u0441\u0442\u044c \u043c\u0430\u0441\u0441\u0438\u0432 (<code style=\"background:#eef0f3;color:#24292f;border-radius:4px;padding:1px 5px;font-size:0.92em;font-family:ui-monospace,SFMono-Regular,Menlo,Consolas,monospace\">np.array<\/code>), \u043d\u043e \u043d\u0435 \u0432\u0441\u044f\u043a\u0438\u0439 \u043c\u0430\u0441\u0441\u0438\u0432 \u0442\u0440\u0430\u043a\u0442\u0443\u044e\u0442 \u043a\u0430\u043a \u0432\u0435\u043a\u0442\u043e\u0440.<\/p>\n<p><strong>\u041c\u043e\u0436\u0435\u0442 \u043b\u0438 \u0434\u043b\u0438\u043d\u0430 \u0432\u0435\u043a\u0442\u043e\u0440\u0430 \u0431\u044b\u0442\u044c \u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439?<\/strong><br \/>\n\u041d\u0435\u0442. \u0414\u043b\u0438\u043d\u0430 (\u043d\u043e\u0440\u043c\u0430) &#8212; \u044d\u0442\u043e \u043a\u043e\u0440\u0435\u043d\u044c \u0438\u0437 \u0441\u0443\u043c\u043c\u044b \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043e\u0432 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442, \u043e\u043d\u0430 \u0432\u0441\u0435\u0433\u0434\u0430 \u043d\u0435\u043e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u0430 \u0438 \u0440\u0430\u0432\u043d\u0430 \u043d\u0443\u043b\u044e \u0442\u043e\u043b\u044c\u043a\u043e \u0443 \u043d\u0443\u043b\u0435\u0432\u043e\u0433\u043e \u0432\u0435\u043a\u0442\u043e\u0440\u0430. \u041e\u0442\u0440\u0438\u0446\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u043c\u0438 \u0431\u044b\u0432\u0430\u044e\u0442 \u043e\u0442\u0434\u0435\u043b\u044c\u043d\u044b\u0435 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u044b, \u043d\u043e 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\u044d\u043c\u0431\u0435\u0434\u0434\u0438\u043d\u0433\u043e\u0432.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0412\u0435\u043a\u0442\u043e\u0440 &#8212; \u044d\u0442\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u043e\u0431\u044a\u0435\u043a\u0442, \u0443 \u043a\u043e\u0442\u043e\u0440\u043e\u0433\u043e \u0435\u0441\u0442\u044c \u0434\u043b\u0438\u043d\u0430 \u0438 \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435. \u0423 \u043d\u0435\u0433\u043e \u0434\u0432\u0430 \u0440\u0430\u0432\u043d\u043e\u043f\u0440\u0430\u0432\u043d\u044b\u0445 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u0438\u044f, \u0438 \u0438\u0445 \u0432\u0430\u0436\u043d\u043e \u043d\u0435 \u043f\u0443\u0442\u0430\u0442\u044c. \u0413\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438 \u0432\u0435\u043a\u0442\u043e\u0440 &#8212; \u044d\u0442\u043e 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[&hellip;]<\/p>\n","protected":false},"author":11,"featured_media":11090,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[578,44],"class_list":["post-11089","post","type-post","status-publish","format-standard","has-post-thumbnail","","category-polza","tag-vektor","tag-matematika"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.3 - aioseo.com -->\n\t<meta name=\"description\" content=\"\u0420\u0430\u0437\u0431\u0438\u0440\u0430\u0435\u043c \u0432\u0435\u043a\u0442\u043e\u0440\u044b \u0432 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0435 \u0438 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u043c\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u0438: \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435, \u0441\u043b\u043e\u0436\u0435\u043d\u0438\u0435, \u0441\u043a\u0430\u043b\u044f\u0440\u043d\u043e\u0435 \u0438 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