{"common":{"save":0,"post_id":"8277","level":3,"total":10,"point":10,"point_extra":0},"segment":[{"id":"6084","post_id":"8277","mon_id":"1159285","chapter_id":"1159382","question":"<p>Trong kh&ocirc;ng gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz cho \u0111\u01b0\u1eddng th\u1eb3ng&nbsp;<span class=\"math-tex\">$d:\\begin{cases}x=1-t\\\\y=2+2t\\\\z=3+t\\end{cases}$<\/span>&nbsp;,<br \/>t &isin; R v&agrave; m\u1eb7t ph\u1eb3ng (P):&nbsp;<span class=\"math-tex\">$x-y+3=0$<\/span>&nbsp;. T&iacute;nh s\u1ed1 \u0111o g&oacute;c gi\u1eefa \u0111\u01b0\u1eddng th\u1eb3ng d v&agrave; m\u1eb7t ph\u1eb3ng (P).<\/p>","options":["<strong>A.<\/strong> 60&deg;","<strong>B.<\/strong> 30&deg;","<strong>C.<\/strong> 120&deg;","<strong>D.<\/strong> 45&deg;"],"correct":"1","level":"3","hint":"","answer":"<p>Ch\u1ecdn&nbsp;<span style=\"color:#16a085;\"><strong>A.<\/strong> 60&deg;.<\/span><\/p><p>\u0110\u01b0\u1eddng th\u1eb3ng d&nbsp;c&oacute; vector&nbsp;ch\u1ec9 ph\u01b0\u01a1ng l&agrave; <span class=\"math-tex\">$\\overrightarrow{u}=(-1;2;1)$<\/span>.<\/p><p>M\u1eb7t ph\u1eb3ng (P)&nbsp;c&oacute; vector&nbsp;ph&aacute;p tuy\u1ebfn l&agrave; <span class=\"math-tex\">$\\overrightarrow{n}=(1;-1;0)$<\/span>.<\/p><p>G\u1ecdi&nbsp;&alpha;&nbsp;l&agrave; g&oacute;c gi\u1eefa \u0111\u01b0\u1eddng th\u1eb3ng d&nbsp;v&agrave; m\u1eb7t ph\u1eb3ng (P).&nbsp;Khi&nbsp;\u0111&oacute; ta c&oacute;<\/p><p><span class=\"math-tex\">$\\sin\\alpha=\\dfrac{|\\overrightarrow{u}.\\overrightarrow{n}|}{|\\overrightarrow{u}|.|\\overrightarrow{n}|} =\\dfrac{|-1.1+2.(-1)+1.0|}{\\sqrt{(-1)^2+2^2+1^2}.\\sqrt{1^2+(-1)^2+0^2}} =\\dfrac{3}{2\\sqrt{3}}=\\dfrac{\\sqrt{3}}{2}$<\/span>.<\/p><p>Suy ra&nbsp;&alpha; = 60&deg;.<\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-12-31 09:02:34","option_type":"txt","len":1},{"id":"6086","post_id":"8277","mon_id":"1159285","chapter_id":"1159382","question":"<p>Trong kh&ocirc;ng gian Oxyz, cho m\u1eb7t ph\u1eb3ng (P):&nbsp;<span class=\"math-tex\">$-\\sqrt{3}x+y+1=0$<\/span>. T&iacute;nh g&oacute;c t\u1ea1o b\u1edfi (P)&nbsp;v\u1edbi tr\u1ee5c Ox.<\/p>","options":["<strong>A.<\/strong> 30&deg;","<strong>B.<\/strong> 60&deg;","<strong>C.<\/strong> 120&deg;","<strong>D.<\/strong> 150&deg;"],"correct":"2","level":"3","hint":"","answer":"<p>Ch\u1ecdn&nbsp;<span style=\"color:#16a085;\"><strong>B.<\/strong> 60&deg;.<\/span><\/p><p>M\u1eb7t ph\u1eb3ng (P) c&oacute; m\u1ed9t vector ph&aacute;p tuy\u1ebfn l&agrave;&nbsp;<span class=\"math-tex\">$\\overrightarrow{n}=(-\\sqrt{3};1;0)$<\/span>.<\/p><p>Tr\u1ee5c Ox c&oacute; m\u1ed9t vector ch\u1ec9 ph\u01b0\u01a1ng l&agrave;&nbsp;<span class=\"math-tex\">$\\overrightarrow{i}=(1;0;0)$<\/span>.<\/p><p>G&oacute;c t\u1ea1o&nbsp;b\u1edfi (P) v\u1edbi tr\u1ee5c Ox l&agrave;<\/p><p><span class=\"math-tex\">$\\sin((P);Ox)=|\\cos((P);Ox)|$<\/span><span class=\"math-tex\">$=\\dfrac{|\\overrightarrow{u}.\\overrightarrow{i}|}{|\\overrightarrow{u}|.|\\overrightarrow{i}|} =\\dfrac{|-\\sqrt{3}.1+1.0+0.0|}{\\sqrt{3+1}.\\sqrt{1}} =\\dfrac{\\sqrt{3}}{2}$<\/span>.<\/p><p>V\u1eady g&oacute;c t\u1ea1o b\u1edfi (P) v&agrave; Ox b\u1eb1ng&nbsp;60&deg;.<\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-12-31 09:07:40","option_type":"txt","len":1},{"id":"6088","post_id":"8277","mon_id":"1159285","chapter_id":"1159382","question":"<p>Cho m\u1eb7t ph\u1eb3ng (P):<span class=\"math-tex\">$3x+4y+5z+2=0$<\/span> v&agrave; \u0111\u01b0\u1eddng&nbsp;th\u1eb3ng d l&agrave; giao tuy\u1ebfn c\u1ee7a hai m\u1eb7t ph\u1eb3ng&nbsp;<span class=\"math-tex\">$(\\alpha):x-2y+1=0$<\/span>;&nbsp;<span class=\"math-tex\">$(\\beta):x-2z-3=0$<\/span>. G\u1ecdi <span class=\"math-tex\">$\\varphi$<\/span>&nbsp;l&agrave; g&oacute;c gi\u1eefa&nbsp;\u0111\u01b0\u1eddng th\u1eb3ng d v&agrave; m\u1eb7t ph\u1eb3ng (P). T&iacute;nh&nbsp;<span class=\"math-tex\">$\\varphi$<\/span>.<\/p>","options":["<strong>A.<\/strong> 30&deg;","<strong>B.<\/strong> 45&deg;","<strong>C.<\/strong> 90&deg;","<strong>D.<\/strong> 60&deg;"],"correct":"4","level":"3","hint":"","answer":"<p>Ch\u1ecdn&nbsp;<span style=\"color:#16a085;\"><strong>D.<\/strong> 60&deg;.<\/span><\/p><p>M\u1eb7t ph\u1eb3ng (P) c&oacute; m\u1ed9t VTPT l&agrave;&nbsp;<span class=\"math-tex\">$\\overrightarrow{n}=(3;4;5)$<\/span>.<\/p><p>\u0110\u01b0\u1eddng th\u1eb3ng d c&oacute; ph\u01b0\u01a1ng tr&igrave;nh:&nbsp;<span class=\"math-tex\">$\\begin{cases}x=2t\\\\y=\\dfrac{1}{2}+t\\\\z=-\\dfrac{3}{2}+t\\end{cases}$<\/span>, t &isin; R.<\/p><p>Suy&nbsp;ra VTCP cu\u0309a d la\u0300&nbsp;<span class=\"math-tex\">$\\overrightarrow{u}_d=(2;1;1)$<\/span>.<\/p><p>Ta c&oacute;&nbsp;<\/p><p><span class=\"math-tex\">$\\sin\\varphi=|\\cos(\\overrightarrow{n};\\overrightarrow{u}_d)|$<\/span><span class=\"math-tex\">$=\\dfrac{|\\overrightarrow{n}.\\overrightarrow{u}_d|}{|\\overrightarrow{n}|.|\\overrightarrow{u}_d|} =\\dfrac{|3.2+4.1+5.1|}{\\sqrt{9+16+25}.\\sqrt{4+1+1}} =\\dfrac{\\sqrt{3}}{2}$<\/span>.<\/p><p>V\u1eady g&oacute;c t\u1ea1o b\u1edfi (P) v&agrave; d l&agrave;&nbsp;<span class=\"math-tex\">$\\varphi$<\/span>&nbsp;=&nbsp;60&deg;.<\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-12-31 09:18:25","option_type":"txt","len":0}]}