Chú ý: Để đảm bảo quyền lợi và bảo vệ tài khoản của mình
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HomeLớp 12Toán lớp 12 - Sách Kết nối tri thứcBài 15. Phương trình đường thẳng trong không gianBài tập nâng cao
{"common":{"save":0,"post_id":"8276","level":3,"total":10,"point":10,"point_extra":0},"segment":[{"id":"6096","post_id":"8276","mon_id":"1159285","chapter_id":"1159382","question":"<p>Trong không gian, cho m\u1eb7t ph\u1eb3ng <span class=\"math-tex\">$(P):x+y-z-4=0$<\/span> và \u0111i\u1ec3m <span class=\"math-tex\">$A(2;-1;3)$<\/span>. G\u1ecdi <span class=\"math-tex\">$\\Delta$<\/span> là \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua A và song song v\u1edbi (P), bi\u1ebft <span class=\"math-tex\">$\\Delta$<\/span> có m\u1ed9t vector ch\u1ec9 ph\u01b0\u01a1ng là <span class=\"math-tex\">$\\overrightarrow{u}=(a;b;c)$<\/span>, \u0111\u1ed3ng th\u1eddi <span class=\"math-tex\">$\\Delta$<\/span> \u0111\u1ed3ng ph\u1eb3ng và không song song v\u1edbi Oz. Tính <span class=\"math-tex\">$\\dfrac{a}{c}$<\/span>.<\/p>","options":["<strong>A.<\/strong> <span class=\"math-tex\">$2$<\/span>","<strong>B.<\/strong> <span class=\"math-tex\">$-2$<\/span>","<strong>C.<\/strong> <span class=\"math-tex\">$-\\dfrac{1}{2}$<\/span>","<strong>D.<\/strong> <span class=\"math-tex\">$\\dfrac{1}{2}$<\/span>"],"correct":"1","level":"3","hint":"","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><strong>A.<\/strong> <span class=\"math-tex\">$2$<\/span>.<\/span><\/p><p>(P) có m\u1ed9t vector pháp tuy\u1ebfn là <span class=\"math-tex\">$\\overrightarrow{n}=(1;1;-1)$<\/span>.<\/p><p><span class=\"math-tex\">$\\Delta$<\/span> \u0111i qua \u0111i\u1ec3m A(2 ; –1 ; 3) và có m\u1ed9t vector ch\u1ec9 ph\u01b0\u01a1ng là <span class=\"math-tex\">$\\overrightarrow{u}=(a;b;c)$<\/span>.<\/p><p>Oz \u0111i qua \u0111i\u1ec3m O(0 ; 0 ; 0) và có m\u1ed9t vector ch\u1ec9 ph\u01b0\u01a1ng là <span class=\"math-tex\">$\\overrightarrow{k}=(0;0;1)$<\/span>.<\/p><p><span class=\"math-tex\">$\\overrightarrow{OA}=(2;-1;3)$<\/span><\/p><p><span class=\"math-tex\">$\\Delta$<\/span> không song song v\u1edbi Oz ⇔ <span class=\"math-tex\">$a:b:c\\ne0:0:1$<\/span>.<\/p><p><span class=\"math-tex\">$\\Delta$<\/span> \u0111\u1ed3ng ph\u1eb3ng v\u1edbi Oz ⇔ 3 vector <span class=\"math-tex\">$\\overrightarrow{u}$<\/span>, <span class=\"math-tex\">$\\overrightarrow{k}$<\/span>, <span class=\"math-tex\">$\\overrightarrow{OA}$<\/span> \u0111\u1ed3ng ph\u1eb3ng ⇔ <span class=\"math-tex\">$[\\overrightarrow{k},\\overrightarrow{OA}].\\overrightarrow{u}=0$<\/span> <br \/>⇔ a + 2b = 0 ⇔ a = –2b.<\/p><p>Do <span class=\"math-tex\">$\\Delta$<\/span> \/\/ (P) ⇒ <span class=\"math-tex\">$\\overrightarrow{u}\\bot\\overrightarrow{n}$<\/span> ⇔ <span class=\"math-tex\">$\\overrightarrow{u}.\\overrightarrow{n}=0$<\/span> ⇔ <span class=\"math-tex\">$a+b-c=0$<\/span> ⇒ <span class=\"math-tex\">$c=-b$<\/span>. Suy ra <span class=\"math-tex\">$\\dfrac{a}{c}=2$<\/span>.<\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-12-31 09:57:25","option_type":"math","len":0},{"id":"6098","post_id":"8276","mon_id":"1159285","chapter_id":"1159382","question":"<p>Trong không gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz, vi\u1ebft ph\u01b0\u01a1ng trình tham s\u1ed1 c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m <span class=\"math-tex\">$M(1;3;-2)$<\/span>, \u0111\u1ed3ng th\u1eddi song song v\u1edbi giao tuy\u1ebfn c\u1ee7a hai m\u1eb7t ph\u1eb3ng <span class=\"math-tex\">$(P):x+y-3=0$<\/span> và <span class=\"math-tex\">$(Q):2x-y+z-3=0$<\/span>.<br \/>.<\/p>","options":["<strong>A.<\/strong> <span class=\"math-tex\">$\\begin{cases}x=1+3t\\\\y=3-t\\\\z=-2+t\\end{cases}$<\/span>","<strong>B.<\/strong> <span class=\"math-tex\">$\\begin{cases}x=1-3t\\\\y=3+t\\\\z=-2+t\\end{cases}$<\/span>","<strong>C.<\/strong> <span class=\"math-tex\">$\\begin{cases}x=1+t\\\\y=3-t\\\\z=-2-3t\\end{cases}$<\/span>","<strong>D.<\/strong> <span class=\"math-tex\">$\\begin{cases}x=1+t\\\\y=3+t\\\\z=-2-3t\\end{cases}$<\/span>"],"correct":"3","level":"3","hint":"","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><strong>C.<\/strong> <span class=\"math-tex\">$\\begin{cases}x=1+t\\\\y=3-t\\\\z=-2-3t\\end{cases}$<\/span>.<\/span><\/p><p>Hai m\u1eb7t ph\u1eb3ng (P) và (Q) có vector pháp tuy\u1ebfn l\u1ea7n l\u01b0\u1ee3t là: <span class=\"math-tex\">$\\overrightarrow{n}_1=(1;1;0),\\overrightarrow{n}_2=(2;-1;1)$<\/span>.<\/p><p>Giao tuy\u1ebfn c\u1ee7a hai m\u1eb7t ph\u1eb3ng (P) và (Q) có vector ch\u1ec9 ph\u01b0\u01a1ng: <span class=\"math-tex\">$\\overrightarrow{u}=[\\overrightarrow{n}_1;\\overrightarrow{n}_2]=(1;-1;-3)$<\/span>.<\/p><p>\u0110\u01b0\u1eddng th\u1eb3ng \u0111i qua \u0111i\u1ec3m M(1 ; 3 ; –2), \u0111\u1ed3ng th\u1eddi song song v\u1edbi giao tuy\u1ebfn c\u1ee7a hai m\u1eb7t ph\u1eb3ng (P) và (Q) nh\u1eadn vector <span class=\"math-tex\">$\\overrightarrow{u}$<\/span> làm vect\u01a1 ch\u1ec9 ph\u01b0\u01a1ng có ph\u01b0\u01a1ng trình tham s\u1ed1 là: <span class=\"math-tex\">$\\begin{cases}x=1+t\\\\y=3-t\\\\z=-2-3t\\end{cases}$<\/span>.<\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-12-31 10:05:59","option_type":"math","len":0},{"id":"6100","post_id":"8276","mon_id":"1159285","chapter_id":"1159382","question":"<p>Trong không gian v\u1edbi h\u1ec7 t\u1ecda \u0111\u1ed9 Oxyz cho hai \u0111\u01b0\u1eddng th\u1eb3ng <span class=\"math-tex\">$d_1:\\dfrac{x-1}{2}=\\dfrac{y}{1}=\\dfrac{z+2}{-2}$<\/span>, <span class=\"math-tex\">$d_2:\\dfrac{x+2}{-2}=\\dfrac{y-1}{-1}=\\dfrac{z}{2}$<\/span>. Xét v\u1ecb trí t\u01b0\u01a1ng \u0111\u1ed1i c\u1ee7a hai \u0111\u01b0\u1eddng th\u1eb3ng \u0111ã cho.<\/p>","options":["<strong>A.<\/strong> Chéo nhau","<strong>B.<\/strong> Trùng nhau","<strong>C.<\/strong> Song song","<strong>D.<\/strong> C\u1eaft nhau"],"correct":"3","level":"3","hint":"","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><strong>C.<\/strong> Song song.<\/span><\/p><p><span class=\"math-tex\">$d_1:\\dfrac{x-1}{2}=\\dfrac{y}{1}=\\dfrac{z+2}{-2}$<\/span> có m\u1ed9t VTCP là <span class=\"math-tex\">$\\overrightarrow{u}_1=(2;1;-2)$<\/span>.<\/p><p><span class=\"math-tex\">$d_2:\\dfrac{x+2}{-2}=\\dfrac{y-1}{-1}=\\dfrac{z}{2}$<\/span> có m\u1ed9t VTCP là <span class=\"math-tex\">$\\overrightarrow{u}_2=(-2;-1;2)$<\/span>.<\/p><p>Vì <span class=\"math-tex\">$\\overrightarrow{u}_1=-\\overrightarrow{u}_2$<\/span> nên <span class=\"math-tex\">$d_1$<\/span> \/\/ <span class=\"math-tex\">$d_2$<\/span> ho\u1eb7c <span class=\"math-tex\">$d_1\\equiv d_2$<\/span>.<\/p><p>\u0110i\u1ec3m M(1 ; 0 ; –2) ∈ <span class=\"math-tex\">$d_1$<\/span> và M(1 ; 0 ; –2) ∉ <span class=\"math-tex\">$d_2$<\/span> nên <span class=\"math-tex\">$d_1$<\/span> \/\/ <span class=\"math-tex\">$d_2$<\/span>.<\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-12-31 10:13:01","option_type":"txt","len":2}]}