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{"common":{"save":0,"post_id":"8208","level":1,"total":10,"point":10,"point_extra":0},"segment":[{"id":"6033","post_id":"8208","mon_id":"1159285","chapter_id":"1159382","question":"<p>Trong không gian Oxyz, t\u1ecda \u0111\u1ed9 m\u1ed9t vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{n}$<\/span> vuông góc v\u1edbi c\u1ea3 hai vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{a}=(1;1;-2)$<\/span>, <span class=\"math-tex\">$\\overrightarrow{b}=(1;0;3)$<\/span> là:<\/p>","options":["<strong>A.<\/strong> <span class=\"math-tex\">$(2;3;-1)$<\/span>","<strong>B.<\/strong> <span class=\"math-tex\">$(3;5;-2)$<\/span>","<strong>C.<\/strong> <span class=\"math-tex\">$(2;-3;-1)$<\/span>","<strong>D.<\/strong> <span class=\"math-tex\">$(3;-5;-1)$<\/span>"],"correct":"4","level":"1","hint":"<p>S\u1eed d\u1ee5ng \u0111\u1ecbnh ngh\u0129a <span class=\"math-tex\">$\\overrightarrow{n}=\\big[\\overrightarrow{a},\\overrightarrow{b}\\big]$<\/span> thì <span class=\"math-tex\">$\\overrightarrow{n}$<\/span> vuông góc v\u1edbi c\u1ea3 hai vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{a}$<\/span> và <span class=\"math-tex\">$\\overrightarrow{b}$<\/span>.<\/p><p>Vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{n}=(a_2b_3-a_3b_2;a_3b_1-a_1b_3;a_1b_2-a_2b_1)$<\/span> \u0111\u01b0\u1ee3c g\u1ecdi là tích có h\u01b0\u1edbng c\u1ee7a hai vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{a}=(a_1;a_2;a_3)$<\/span> và <span class=\"math-tex\">$\\overrightarrow{b}=(b_1;b_2;b_3)$<\/span>, kí hi\u1ec7u là <span class=\"math-tex\">$\\big[\\overrightarrow{a},\\overrightarrow{b}\\big]$<\/span>.<\/p><p><span class=\"math-tex\">$\\big[\\overrightarrow{a},\\overrightarrow{b}\\big]=$<\/span><span class=\"math-tex\">$\\bigg(\\bigg|\\begin{matrix}\\n a_2 & a_3 \\\\\\n b_2 & b_3\\n \\end{matrix}\\bigg|;\\bigg|\\begin{matrix}\\n a_3 & a_1 \\\\\\n b_3 & b_1\\n \\end{matrix}\\bigg|;\\bigg|\\begin{matrix}\\n a_1 & a_2 \\\\\\n b_1 & b_2\\n \\end{matrix}\\bigg|\\bigg)$<\/span><span class=\"math-tex\">$=(a_2b_3-a_3b_2;a_3b_1-a_1b_3;a_1b_2-a_2b_1)$<\/span><\/p>","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><strong>D.<\/strong> <span class=\"math-tex\">$(3;-5;-1)$<\/span><\/span><\/p><p>Ta có: <span class=\"math-tex\">$\\big[\\overrightarrow{a},\\overrightarrow{b}\\big]=\\Big(1.3-0.(-2);(-2).1-3.1;1.0-1.1\\Big)=(3;-5;-1)$<\/span><\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-12-25 08:04:10","option_type":"math","len":0},{"id":"6035","post_id":"8208","mon_id":"1159285","chapter_id":"1159382","question":"<p>Trong không gian Oxyz, cho hai vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{a}=(2;1;-2)$<\/span>, <span class=\"math-tex\">$\\overrightarrow{b}=(1;0;2)$<\/span>. Tìm t\u1ecda \u0111\u1ed9 vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{c}$<\/span> là tích có h\u01b0\u1edbng c\u1ee7a <span class=\"math-tex\">$\\overrightarrow{a}$<\/span> và <span class=\"math-tex\">$\\overrightarrow{b}$<\/span>.<\/p>","options":["<strong>A.<\/strong> <span class=\"math-tex\">$\\overrightarrow{c}=(2;6;-1)$<\/span>","<strong>B.<\/strong> <span class=\"math-tex\">$\\overrightarrow{c}=(4;6;-1)$<\/span>","<strong>C.<\/strong> <span class=\"math-tex\">$\\overrightarrow{c}=(2;-6;-1)$<\/span>","<strong>D.<\/strong> <span class=\"math-tex\">$\\overrightarrow{c}=(4;-6;-1)$<\/span>"],"correct":"3","level":"1","hint":"<p>S\u1eed d\u1ee5ng \u0111\u1ecbnh ngh\u0129a <span class=\"math-tex\">$\\overrightarrow{n}=\\big[\\overrightarrow{a},\\overrightarrow{b}\\big]$<\/span> thì <span class=\"math-tex\">$\\overrightarrow{n}$<\/span> vuông góc v\u1edbi c\u1ea3 hai vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{a}$<\/span> và <span class=\"math-tex\">$\\overrightarrow{b}$<\/span>.<\/p><p>Vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{n}=(a_2b_3-a_3b_2;a_3b_1-a_1b_3;a_1b_2-a_2b_1)$<\/span> \u0111\u01b0\u1ee3c g\u1ecdi là tích có h\u01b0\u1edbng c\u1ee7a hai vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{a}=(a_1;a_2;a_3)$<\/span> và <span class=\"math-tex\">$\\overrightarrow{b}=(b_1;b_2;b_3)$<\/span>, kí hi\u1ec7u là <span class=\"math-tex\">$\\big[\\overrightarrow{a},\\overrightarrow{b}\\big]$<\/span>.<\/p><p><span class=\"math-tex\">$\\big[\\overrightarrow{a},\\overrightarrow{b}\\big]=$<\/span><span class=\"math-tex\">$\\bigg(\\bigg|\\begin{matrix} a_2 & a_3 \\\\ b_2 & b_3 \\end{matrix}\\bigg|;\\bigg|\\begin{matrix} a_3 & a_1 \\\\ b_3 & b_1 \\end{matrix}\\bigg|;\\bigg|\\begin{matrix} a_1 & a_2 \\\\ b_1 & b_2 \\end{matrix}\\bigg|\\bigg)$<\/span><span class=\"math-tex\">$=(a_2b_3-a_3b_2;a_3b_1-a_1b_3;a_1b_2-a_2b_1)$<\/span><\/p>","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><strong>C.<\/strong> <span class=\"math-tex\">$\\overrightarrow{c}=(2;-6;-1)$<\/span><\/span><\/p><p>Áp d\u1ee5ng công th\u1ee9c tính tích có h\u01b0\u1edbng trong h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 Oxyz ta \u0111\u01b0\u1ee3c:<\/p><p><span class=\"math-tex\">$\\overrightarrow{c}=\\Big[\\overrightarrow{a},\\overrightarrow{b}\\Big]=(2;-6;-1)$<\/span><\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-12-25 08:08:30","option_type":"math","len":0},{"id":"6036","post_id":"8208","mon_id":"1159285","chapter_id":"1159382","question":"<p>Trong không gian v\u1edbi h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 Oxyz , cho <span class=\"math-tex\">$A(2;1;-3)$<\/span>, <span class=\"math-tex\">$B(0;-2;5)$<\/span> và <span class=\"math-tex\">$C(1;1;3)$<\/span>. Tìm t\u1ecda \u0111\u1ed9 vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{n}$<\/span> có ph\u01b0\u01a1ng vuông góc v\u1edbi hai vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{AB}$<\/span> và <span class=\"math-tex\">$\\overrightarrow{AC}$<\/span>.<\/p>","options":["<strong>A.<\/strong> <span class=\"math-tex\">$\\overrightarrow{n}=(-18;4;-3)$<\/span>","<strong>B.<\/strong> <span class=\"math-tex\">$\\overrightarrow{n}=(-18;0;-3)$<\/span>","<strong>C.<\/strong> <span class=\"math-tex\">$\\overrightarrow{n}=(8;4;-3)$<\/span>","<strong>D.<\/strong> <span class=\"math-tex\">$\\overrightarrow{n}=(1;4;-3)$<\/span>"],"correct":"1","level":"1","hint":"<p>S\u1eed d\u1ee5ng \u0111\u1ecbnh ngh\u0129a <span class=\"math-tex\">$\\overrightarrow{n}=\\big[\\overrightarrow{a},\\overrightarrow{b}\\big]$<\/span> thì <span class=\"math-tex\">$\\overrightarrow{n}$<\/span> vuông góc v\u1edbi c\u1ea3 hai vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{a}$<\/span> và <span class=\"math-tex\">$\\overrightarrow{b}$<\/span>.<\/p><p>Vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{n}=(a_2b_3-a_3b_2;a_3b_1-a_1b_3;a_1b_2-a_2b_1)$<\/span> \u0111\u01b0\u1ee3c g\u1ecdi là tích có h\u01b0\u1edbng c\u1ee7a hai vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{a}=(a_1;a_2;a_3)$<\/span> và <span class=\"math-tex\">$\\overrightarrow{b}=(b_1;b_2;b_3)$<\/span>, kí hi\u1ec7u là <span class=\"math-tex\">$\\big[\\overrightarrow{a},\\overrightarrow{b}\\big]$<\/span>.<\/p><p><span class=\"math-tex\">$\\big[\\overrightarrow{a},\\overrightarrow{b}\\big]=$<\/span><span class=\"math-tex\">$\\bigg(\\bigg|\\begin{matrix} a_2 & a_3 \\\\ b_2 & b_3 \\end{matrix}\\bigg|;\\bigg|\\begin{matrix} a_3 & a_1 \\\\ b_3 & b_1 \\end{matrix}\\bigg|;\\bigg|\\begin{matrix} a_1 & a_2 \\\\ b_1 & b_2 \\end{matrix}\\bigg|\\bigg)$<\/span><span class=\"math-tex\">$=(a_2b_3-a_3b_2;a_3b_1-a_1b_3;a_1b_2-a_2b_1)$<\/span><\/p>","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><strong>A.<\/strong> <span class=\"math-tex\">$\\overrightarrow{n}=(-18;4;-3)$<\/span><\/span><\/p><p>Ta có: <span class=\"math-tex\">$\\overrightarrow{AB}=(-2;-3;8)$<\/span> và <span class=\"math-tex\">$\\overrightarrow{AC}=(-1;0;6)$<\/span>.<\/p><p>Suy ra <span class=\"math-tex\">$\\big[\\overrightarrow{AB},\\overrightarrow{AC}\\big]=(-18;4;-3)$<\/span>.<\/p><p>V\u1eady <span class=\"math-tex\">$\\overrightarrow{n}=\\big[\\overrightarrow{AB},\\overrightarrow{AC}\\big]=(-18;4;-3)$<\/span>.<\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-12-25 08:14:52","option_type":"math","len":0}]}