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HomeLớp 12Toán lớp 12 - Sách Kết nối tri thứcBài tập cuối chương 5 - PHƯƠNG PHÁP TOẠ ĐỘ TRONG KHÔNG GIANBài tập trung bình
{"common":{"save":0,"post_id":"8736","level":2,"total":10,"point":10,"point_extra":0},"segment":[{"id":"6304","post_id":"8736","mon_id":"1159285","chapter_id":"1159382","question":"<p>Trong không gian Oxyz cho \u0111\u01b0\u1eddng th\u1eb3ng <span class=\"math-tex\">$\\Delta:\\dfrac{x-2024}{2}=\\dfrac{y}{1}=\\dfrac{z+2025}{-2}$<\/span> và m\u1eb7t ph\u1eb3ng <span class=\"math-tex\">$(P):2x+2y-z+1=0$<\/span>. Tính góc gi\u1eefa \u0111\u01b0\u1eddng th\u1eb3ng <span class=\"math-tex\">$\\Delta$<\/span> và m\u1eb7t ph\u1eb3ng (P) (làm tròn \u0111\u1ebfn hàng \u0111\u01a1n v\u1ecb c\u1ee7a \u0111\u1ed9).<\/p>","options":["<strong>A.<\/strong> 64°","<strong>B.<\/strong> 63°","<strong>C.<\/strong> 61°","<strong>D.<\/strong> 62°"],"correct":"2","level":"2","hint":"","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><strong>B.<\/strong> 63°.<\/span><\/p><p>Ta có: <span class=\"math-tex\">$\\overrightarrow{u}=(2;1;-2)$<\/span> là m\u1ed9t vector ch\u1ec9 ph\u01b0\u01a1ng c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng <span class=\"math-tex\">$\\Delta$<\/span> và <span class=\"math-tex\">$\\overrightarrow{n}=(2;2;-1)$<\/span> là m\u1ed9t vector pháp tuy\u1ebfn c\u1ee7a m\u1eb7t ph\u1eb3ng (P).<\/p><p>Khi \u0111ó, ta có<\/p><p> <span class=\"math-tex\">$\\sin(\\Delta,(P))=|\\cos(\\overrightarrow{u},\\overrightarrow{n})|=\\dfrac{|\\overrightarrow{u}.\\overrightarrow{n}|}{|\\overrightarrow{u}|.|\\overrightarrow{n}|}=\\dfrac{|2.2+1.2+(-2).(-1)|}{\\sqrt{4+1+4}.\\sqrt{4+4+1}}=\\dfrac{8}{9}$<\/span>.<\/p><p>Suy ra <span class=\"math-tex\">$\\cos(\\Delta,(P))=\\sqrt{1-\\sin^2(\\Delta,(P))}=\\sqrt{1-\\bigg(\\dfrac{8}{9}\\bigg)^2}=\\dfrac{\\sqrt{17}}{9}$<\/span>.<\/p><p>V\u1eady góc gi\u1eefa \u0111\u01b0\u1eddng th\u1eb3ng <span class=\"math-tex\">$\\Delta$<\/span> và m\u1eb7t ph\u1eb3ng (P) b\u1eb1ng kho\u1ea3ng 63°.<\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2025-04-03 09:22:28","option_type":"txt","len":0},{"id":"6305","post_id":"8736","mon_id":"1159285","chapter_id":"1159382","question":"<p>Trong không gian v\u1edbi h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 Oxyz, tính góc gi\u1eefa m\u1eb7t ph\u1eb3ng (Oxy) và m\u1eb7t ph\u1eb3ng (P): x + z + 1 = 0.<\/p>","options":["<strong>A.<\/strong> 30°","<strong>B.<\/strong> 45”","<strong>C.<\/strong> 60°","<strong>D.<\/strong> 90°"],"correct":"2","level":"2","hint":"","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><strong>B.<\/strong> 45”.<\/span><\/p><p><span style=\"color:#16a085;\"><\/span>M\u1eb7t ph\u1eb3ng (Oxy) có m\u1ed9t vector pháp tuy\u1ebfn là <span class=\"math-tex\">$\\overrightarrow{k}=(0;0;1)$<\/span>.<br \/>M\u1eb7t ph\u1eb3ng (P) có m\u1ed9t vector pháp tuy\u1ebfn là <span class=\"math-tex\">$\\overrightarrow{n}=(1;0;1)$<\/span>.<br \/>G\u1ecdi α là góc gi\u1eefa m\u1eb7t ph\u1eb3ng (Oxy) và m\u1eb7t ph\u1eb3ng (P), khi \u0111ó<\/p><p><span class=\"math-tex\">$\\cos\\alpha=\\dfrac{|\\overrightarrow{k}.\\overrightarrow{n}|}{|\\overrightarrow{k}|.|\\overrightarrow{n}|}=\\dfrac{|0.1+0.0+1.1|}{\\sqrt{0+0+1}.\\sqrt{1+0+1}}=\\dfrac{1}{\\sqrt{2}}$<\/span>.<\/p><p>Suy ra α = 45°.<\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2025-04-03 09:29:10","option_type":"txt","len":1},{"id":"6308","post_id":"8736","mon_id":"1159285","chapter_id":"1159382","question":"<p>Trong không gian v\u1edbi h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 Oxyz, cho hai \u0111\u01b0\u1eddng th\u1eb3ng có ph\u01b0\u01a1ng trình <span class=\"math-tex\">$d_1:\\dfrac{x+1}{1}=\\dfrac{y-2}{1}=\\dfrac{z+2}{-1}$<\/span> và <span class=\"math-tex\">$d_2:\\dfrac{x-1}{1}=\\dfrac{y+2}{-1}=\\dfrac{z+1}{1}$<\/span>. Khi \u0111ó cosin c\u1ee7a góc gi\u1eefa hai \u0111\u01b0\u1eddng th\u1eb3ng <span class=\"math-tex\">$d_1$<\/span> và <span class=\"math-tex\">$d_2$<\/span> là<\/p>","options":["<strong>A.<\/strong> <span class=\"math-tex\">$\\dfrac{1}{3}$<\/span>","<strong>B.<\/strong> <span class=\"math-tex\">$\\dfrac{1}{2}$<\/span>","<strong>C.<\/strong> <span class=\"math-tex\">$\\dfrac{1}{\\sqrt{3}}$<\/span>","<strong>D.<\/strong> <span class=\"math-tex\">$\\dfrac{1}{\\sqrt{2}}$<\/span>"],"correct":"1","level":"2","hint":"","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><strong>A.<\/strong> <span class=\"math-tex\">$\\dfrac{1}{3}$<\/span>.<\/span><\/p><p><span style=\"color:#16a085;\"><\/span>\u0110\u01b0\u1eddng th\u1eb3ng <span class=\"math-tex\">$d_1$<\/span> có vector ch\u1ec9 ph\u01b0\u01a1ng <span class=\"math-tex\">$\\overrightarrow{u_1}=(1;1;-1)$<\/span>.<\/p><p>\u0110\u01b0\u1eddng th\u1eb3ng <span class=\"math-tex\">$d_2$<\/span> có vector ch\u1ec9 ph\u01b0\u01a1ng <span class=\"math-tex\">$\\overrightarrow{u_2}=(1;-1;1)$<\/span>.<\/p><p>G\u1ecdi α là góc gi\u1eefa \u0111\u01b0\u1eddng th\u1eb3ng <span class=\"math-tex\">$d_1$<\/span> và \u0111\u01b0\u1eddng th\u1eb3ng <span class=\"math-tex\">$d_2$<\/span>, khi \u0111ó<\/p><p><span class=\"math-tex\">$\\cos\\alpha=\\dfrac{|\\overrightarrow{u_1}.\\overrightarrow{u_2}|}{|\\overrightarrow{u_1}|.|\\overrightarrow{u_2}|}=\\dfrac{|1.1-1.1-1.1|}{\\sqrt{1+1+1}.\\sqrt{1+1+1}}=\\dfrac{1}{3}$<\/span>.<\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2025-04-03 09:38:04","option_type":"math","len":0}]}