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HomeLớp 12Toán lớp 12 - Sách Kết nối tri thứcBài 15. Phương trình đường thẳng trong không gianBài tập trung bình
{"common":{"save":0,"post_id":"8275","level":2,"total":10,"point":10,"point_extra":0},"segment":[{"id":"6099","post_id":"8275","mon_id":"1159285","chapter_id":"1159382","question":"<p>Trong không gian Oxyz, cho \u0111i\u1ec3m A(–4; –3; 3) và m\u1eb7t ph\u1eb3ng <span class=\"math-tex\">$(P):x+y+z=0$<\/span>. \u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua A, c\u1eaft tr\u1ee5c Oz<br \/>và song song v\u1edbi (P) có ph\u01b0\u01a1ng trình là <\/p>","options":["<strong>A.<\/strong> <span class=\"math-tex\">$\\dfrac{x+8}{4}=\\dfrac{y+6}{3}=\\dfrac{z-10}{-7}$<\/span>","<strong>B.<\/strong> <span class=\"math-tex\">$\\dfrac{x+4}{4}=\\dfrac{y+3}{3}=\\dfrac{z-3}{1}$<\/span>","<strong>C.<\/strong> <span class=\"math-tex\">$\\dfrac{x+4}{-4}=\\dfrac{y+3}{3}=\\dfrac{z-3}{1}$<\/span>","<strong>D.<\/strong> <span class=\"math-tex\">$\\dfrac{x-4}{4}=\\dfrac{y-3}{3}=\\dfrac{z-3}{-7}$<\/span>"],"correct":"1","level":"2","hint":"<p>G\u1ecdi d là \u0111\u01b0\u1eddng th\u1eb3ng c\u1ea7n l\u1eadp.<\/p><p>+) d c\u1eaft tr\u1ee5c Oz t\u1ea1i \u0111i\u1ec3m B, d \u0111i qua A nên <span class=\"math-tex\">$\\overrightarrow{AB}$<\/span> là 1 VTCP c\u1ee7a d.<\/p><p>+) d song song v\u1edbi (P) nên <span class=\"math-tex\">$\\overrightarrow{AB}\\;\\bot\\;\\overrightarrow{n}$<\/span> ⇔ <span class=\"math-tex\">$\\overrightarrow{AB}\\;.\\;\\overrightarrow{n}=0$<\/span>. Suy ra to\u1ea1 \u0111\u1ed9 vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{AB}$<\/span>.<\/p><p>+) Vi\u1ebft ph\u01b0\u01a1ng trình \u0111\u01b0\u1eddng th\u1eb3ng d<\/p>","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><strong>A.<\/strong> <span class=\"math-tex\">$\\dfrac{x+8}{4}=\\dfrac{y+6}{3}=\\dfrac{z-10}{-7}$<\/span><\/span><\/p><p>G\u1ecdi d là \u0111\u01b0\u1eddng th\u1eb3ng c\u1ea7n l\u1eadp.<\/p><p>M\u1eb7t ph\u1eb3ng (P) có m\u1ed9t VTPT <span class=\"math-tex\">$\\overrightarrow{n}=(1;1;1)$<\/span>.<\/p><p>Theo \u0111\u1ec1 ta có d ∩ Oz t\u1ea1i B(0; 0; c) ⇒ <span class=\"math-tex\">$\\overrightarrow{AB}=(4;3;c-3)$<\/span> là m\u1ed9t VTCP c\u1ee7a d.<\/p><p>Khi \u0111ó <span class=\"math-tex\">$\\overrightarrow{AB}\\;\\bot\\;\\overrightarrow{n}$<\/span> ⇔ <span class=\"math-tex\">$\\overrightarrow{AB}\\;.\\;\\overrightarrow{n}=0$<\/span> ⇔ <span class=\"math-tex\">$4.1+3.1+(c-3).1=0$<\/span> ⇔ <span class=\"math-tex\">$c-3=-7$<\/span>.<\/p><p>Suy ra <span class=\"math-tex\">$\\overrightarrow{AB}=(4;3;-7)$<\/span>.<\/p><p>V\u1eady <span class=\"math-tex\">$d:\\dfrac{x+4}{4}=\\dfrac{y+3}{3}=\\dfrac{z-3}{-7}$<\/span> hay <span class=\"math-tex\">$d:\\dfrac{x+8}{4}=\\dfrac{y+6}{3}=\\dfrac{z-10}{-7}$<\/span><\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-12-31 10:06:22","option_type":"math","len":0},{"id":"6102","post_id":"8275","mon_id":"1159285","chapter_id":"1159382","question":"<p>Trong không gian v\u1edbi h\u1ec7 to\u1ea1 \u0111\u1ed9 Oxyz, cho \u0111i\u1ec3m <span class=\"math-tex\">$M(1;-3;4)$<\/span>, \u0111\u01b0\u1eddng th\u1eb3ng <span class=\"math-tex\">$d:\\dfrac{x+2}{3}=\\dfrac{y-5}{-5}=\\dfrac{z-2}{-1}$<\/span> và m\u1eb7t ph\u1eb3ng <span class=\"math-tex\">$(P):2x+z-2=0$<\/span>. Vi\u1ebft ph\u01b0\u01a1ng trình \u0111\u01b0\u1eddng th\u1eb3ng <span class=\"math-tex\">$\\Delta$<\/span> qua M và vuông góc v\u1edbi d và song song v\u1edbi (P).<\/p>","options":["<strong>A.<\/strong> <span class=\"math-tex\">$\\Delta:\\dfrac{x-1}{1}=\\dfrac{y+3}{-1}=\\dfrac{z-4}{-2}$<\/span>","<strong>B.<\/strong> <span class=\"math-tex\">$\\Delta:\\dfrac{x-1}{-1}=\\dfrac{y+3}{-1}=\\dfrac{z-4}{-2}$<\/span>","<strong>C.<\/strong> <span class=\"math-tex\">$\\Delta:\\dfrac{x-1}{1}=\\dfrac{y+3}{1}=\\dfrac{z-4}{-2}$<\/span>","<strong>D.<\/strong> <span class=\"math-tex\">$\\Delta:\\dfrac{x-1}{1}=\\dfrac{y+3}{-1}=\\dfrac{z-4}{2}$<\/span>"],"correct":"3","level":"2","hint":"<p>Ta có: <span class=\"math-tex\">$\\overrightarrow{u_d}=(3;-5;-1)$<\/span> là véc t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a d.<\/p><p><span class=\"math-tex\">$\\overrightarrow{n_{_P}}=(2;0;1)$<\/span> là véc t\u01a1 pháp tuy\u1ebfn c\u1ee7a (P).<\/p><p>⇒ <span class=\"math-tex\">$\\Big[\\overrightarrow{u_{_d}},\\overrightarrow{n_{_P}}\\Big]=(-5;-5;10)$<\/span> ⇒ <span class=\"math-tex\">$\\overrightarrow{u}=-\\dfrac{1}{5}\\Big[\\overrightarrow{u_{_d}},\\overrightarrow{n_{_P}}\\Big]=(1;1;-2)$<\/span><\/p><p>Do <span class=\"math-tex\">$\\Delta$<\/span> vuông góc v\u1edbi d và song song v\u1edbi (P) nên <span class=\"math-tex\">$\\overrightarrow{u}=(1;1;-2)$<\/span> là véct\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a <span class=\"math-tex\">$\\Delta$<\/span>.<\/p>","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><strong>C.<\/strong> <span class=\"math-tex\">$\\Delta:\\dfrac{x-1}{1}=\\dfrac{y+3}{1}=\\dfrac{z-4}{-2}$<\/span><\/span><\/p><p>Ta có: <span class=\"math-tex\">$\\overrightarrow{u_d}=(3;-5;-1)$<\/span> là véc t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a d.<\/p><p><span class=\"math-tex\">$\\overrightarrow{n_{_P}}=(2;0;1)$<\/span> là véc t\u01a1 pháp tuy\u1ebfn c\u1ee7a (P).<\/p><p>⇒ <span class=\"math-tex\">$\\Big[\\overrightarrow{u_{_d}},\\overrightarrow{n_{_P}}\\Big]=(-5;-5;10)$<\/span> ⇒ <span class=\"math-tex\">$\\overrightarrow{u}=-\\dfrac{1}{5}\\Big[\\overrightarrow{u_{_d}},\\overrightarrow{n_{_P}}\\Big]=(1;1;-2)$<\/span><\/p><p>Do <span class=\"math-tex\">$\\Delta$<\/span> vuông góc v\u1edbi d và song song v\u1edbi (P) nên <span class=\"math-tex\">$\\overrightarrow{u}=(1;1;-2)$<\/span> là véct\u01a1 ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a <span class=\"math-tex\">$\\Delta$<\/span>.<\/p><p>Khi \u0111ó ph\u01b0\u01a1ng trình \u0111\u01b0\u1eddng th\u1eb3ng <span class=\"math-tex\">$\\Delta$<\/span> qua <span class=\"math-tex\">$M(1;-3;4)$<\/span> và có véct\u01a1 ch\u1ec9 ph\u01b0\u01a1ng <span class=\"math-tex\">$\\overrightarrow{u}=(1;1;-2)$<\/span> là:<\/p><p><span class=\"math-tex\">$\\Delta:\\dfrac{x-1}{1}=\\dfrac{y+3}{1}=\\dfrac{z-4}{-2}$<\/span><\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2025-01-03 07:46:43","option_type":"math","len":0},{"id":"6103","post_id":"8275","mon_id":"1159285","chapter_id":"1159382","question":"<p>Trong không gian v\u1edbi h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 Oxyz, cho hai m\u1eb7t ph\u1eb3ng <span class=\"math-tex\">$(\\alpha):x-2y+z-1=0\\;$<\/span>, <span class=\"math-tex\">$(\\beta):2x+y-z=0$<\/span> và \u0111i\u1ec3m <span class=\"math-tex\">$A(1;2;-1)$<\/span>. \u0110\u01b0\u1eddng th\u1eb3ng d \u0111i qua \u0111i\u1ec3m A và song song v\u1edbi c\u1ea3 hai m\u1eb7t ph\u1eb3ng (α) và (β) có ph\u01b0\u01a1ng trình là:<\/p>","options":["<strong>A.<\/strong> <span class=\"math-tex\">$d:\\dfrac{x-1}{-2}=\\dfrac{y-2}{4}=\\dfrac{z+1}{-2}$<\/span>","<strong>B.<\/strong> <span class=\"math-tex\">$d:\\dfrac{x-1}{1}=\\dfrac{y-2}{3}=\\dfrac{z+1}{5}$<\/span>","<strong>C.<\/strong> <span class=\"math-tex\">$d:\\dfrac{x-1}{1}=\\dfrac{y-2}{-2}=\\dfrac{z+1}{-1}$<\/span>","<strong>D.<\/strong> <span class=\"math-tex\">$d:\\dfrac{x}{1}=\\dfrac{y+2}{2}=\\dfrac{z-3}{1}$<\/span>"],"correct":"2","level":"2","hint":"","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><strong>B.<\/strong> <span class=\"math-tex\">$d:\\dfrac{x-1}{1}=\\dfrac{y-2}{3}=\\dfrac{z+1}{5}$<\/span><\/span><\/p><p>M\u1eb7t ph\u1eb3ng (α) có véc t\u01a1 pháp tuy\u1ebfn là <span class=\"math-tex\">$\\overrightarrow{n_1}=(1;-2;1)$<\/span>, mp (β) véc t\u01a1 pháp tuy\u1ebfn là <span class=\"math-tex\">$\\overrightarrow{n_2}=(2;1;-1)$<\/span>.<\/p><p>\u0110\u01b0\u1eddng th\u1eb3ng d có véc t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng là <span class=\"math-tex\">$\\overrightarrow{u}=\\Big[\\overrightarrow{n_1},\\overrightarrow{n_{2}}\\Big]=(1;3;5)$<\/span>.<\/p><p>Ph\u01b0\u01a1ng trình \u0111\u01b0\u1eddng th\u1eb3ng d \u0111i qua \u0111i\u1ec3m A và có véc t\u01a1 ch\u1ec9 ph\u01b0\u01a1ng là <span class=\"math-tex\">$\\overrightarrow{u}=(1;3;5)$<\/span> là:<\/p><p><span class=\"math-tex\">$d:\\dfrac{x-1}{1}=\\dfrac{y-2}{3}=\\dfrac{z+1}{5}$<\/span>.<\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2025-01-03 08:02:14","option_type":"math","len":0}]}