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{"common":{"save":0,"post_id":"6458","level":2,"total":10,"point":10,"point_extra":0},"segment":[{"id":"6960","mon_id":"1158767","chapter_id":"1158890","post_id":"6458","question":"<p>Cho hai vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{a}$<\/span> và <span class=\"math-tex\">$\\overrightarrow{b}$<\/span> th\u1ecfa mãn <span class=\"math-tex\">$|\\overrightarrow{a}|=|\\overrightarrow{b}|$<\/span> = 1 và hai vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{u}=2\\overrightarrow{a}-\\dfrac{5}{2}\\overrightarrow{b}$<\/span> và <span class=\"math-tex\">$\\overrightarrow{v}=2\\overrightarrow{a}+\\overrightarrow{b}$<\/span> vuông góc v\u1edbi nhau. Tính s\u1ed1 \u0111o góc gi\u1eefa hai vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{a}$<\/span> và <span class=\"math-tex\">$\\overrightarrow{b}$<\/span> .<\/p>","options":["<strong>A.<\/strong> 90°","<strong>B.<\/strong> 180°","<strong>C.<\/strong> 60°","<strong>D.<\/strong> 45°"],"correct":"3","level":"2","hint":"","answer":"<p>Hai vect\u01a1 vuông góc v\u1edbi nhau khi và ch\u1ec9 khi tích vô h\u01b0\u1edbng c\u1ee7a chúng b\u1eb1ng 0.<\/p><p>Theo gi\u1ea3 thi\u1ebft <span class=\"math-tex\">$\\overrightarrow{u}$<\/span> và <span class=\"math-tex\">$\\overrightarrow{v}$<\/span> vuông góc v\u1edbi nhau nên <span class=\"math-tex\">$\\overrightarrow{u}.\\overrightarrow{v}$<\/span> = 0<\/p><p>⇔ <span class=\"math-tex\">$\\bigg(2\\overrightarrow{a}-\\dfrac{5}{2}\\overrightarrow{b}\\bigg) . \\bigg(2\\overrightarrow{a}+\\overrightarrow{b}\\bigg)$<\/span> = 0<\/p><p>⇔ <span class=\"math-tex\">$4\\bigg(\\overrightarrow{a}\\bigg)^2-\\dfrac{5}{2}\\bigg(\\overrightarrow{b}\\bigg)^2-5\\overrightarrow{a}.\\overrightarrow{b}+2\\overrightarrow{a}.\\overrightarrow{b}$<\/span> = 0<\/p><p>⇔ <span class=\"math-tex\">$4\\bigg(\\overrightarrow{a}\\bigg)^2-\\dfrac{5}{2}\\bigg(\\overrightarrow{b}\\bigg)^2-3\\overrightarrow{a}.\\overrightarrow{b}$<\/span> = 0<\/p><p>Mà <span class=\"math-tex\">$4\\bigg(\\overrightarrow{a}\\bigg)^2 = 4|\\overrightarrow{a}|^2$<\/span> = 4 . 1² = 4; <span class=\"math-tex\">$-\\dfrac{5}{2}\\bigg(\\overrightarrow{b}\\bigg)^2 = -\\dfrac{5}{2}|\\overrightarrow{b}|^2$<\/span> = <span class=\"math-tex\">$-\\dfrac{5}{2}$<\/span> . 1² = <span class=\"math-tex\">$-\\dfrac{5}{2}$<\/span> .<\/p><p>Suy ra 4 <span class=\"math-tex\">$-\\dfrac{5}{2} - 3\\overrightarrow{a}.\\overrightarrow{b}$<\/span> = 0 ⇔ <span class=\"math-tex\">$\\dfrac{3}{2}-3\\overrightarrow{a}.\\overrightarrow{b}$<\/span> = 0 ⇔ <span class=\"math-tex\">$\\overrightarrow{a} . \\overrightarrow{b}$<\/span> = <span class=\"math-tex\">$\\dfrac{1}{2}$<\/span> .<\/p><p>cos<span class=\"math-tex\">$\\big(\\overrightarrow{a},\\overrightarrow{b}\\big) = \\dfrac{\\overrightarrow{a}.\\overrightarrow{b}}{|\\overrightarrow{a}|.|\\overrightarrow{b}|}=\\dfrac{\\dfrac{1}{2}}{1.1}=\\dfrac{1}{2}$<\/span> ⇒ <span class=\"math-tex\">$\\big(\\overrightarrow{a},\\overrightarrow{b}\\big)$<\/span> = 60°.<\/p><p>\u0110áp án \u0111úng là <span style=\"color:#16a085;\"><strong>C.<\/strong> 60°.<\/span><\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2023-05-31 02:55:19","option_type":"txt","len":1},{"id":"7007","mon_id":"1158767","chapter_id":"1158890","post_id":"6458","question":"<p>Trong m\u1eb7t ph\u1eb3ng t\u1ecda \u0111\u1ed9 Oxy, cho hai vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{a}$<\/span> = (3; 4) và <span class=\"math-tex\">$\\overrightarrow{b}$<\/span> = (1; –6).<\/p><p>Tìm t\u1ecda \u0111\u1ed9 vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{c}$<\/span> bi\u1ebft <span class=\"math-tex\">$\\overrightarrow{c}.\\overrightarrow{a}$<\/span> = 11 và <span class=\"math-tex\">$\\overrightarrow{c}.\\overrightarrow{b}$<\/span> = –11.<\/p>","options":["<strong>A.<\/strong> <span class=\"math-tex\">$\\overrightarrow{c}$<\/span> = (1; 2)","<strong>B.<\/strong> <span class=\"math-tex\">$\\overrightarrow{c}$<\/span> = (–1; 2)","<strong>C.<\/strong> <span class=\"math-tex\">$\\overrightarrow{c}$<\/span> = (1; –2)","<strong>D.<\/strong> <span class=\"math-tex\">$\\overrightarrow{c}$<\/span> = (–1; –2)"],"correct":"1","level":"2","hint":"","answer":"<p>G\u1ecdi C(x; y).<\/p><p>Theo gi\u1ea3 thi\u1ebft, <span class=\"math-tex\">$\\overrightarrow{c}.\\overrightarrow{a}$<\/span> = 11 ⇒ 3x + 4y = 11 và <span class=\"math-tex\">$\\overrightarrow{c}.\\overrightarrow{b}$<\/span> = –11 ⇒ x – 6y = –11.<\/p><p>V\u1eady ta có h\u1ec7 <span class=\"math-tex\">$\\begin{cases}3x+4y=11\\\\x-6y=-11\\end{cases}$<\/span> ⇔ <span class=\"math-tex\">$\\begin{cases}x=1\\\\y=2\\end{cases}$<\/span> . Suy ra <span class=\"math-tex\">$\\overrightarrow{c}$<\/span> = (1; 2).<\/p><p>\u0110áp án \u0111úng là <span style=\"color:#16a085;\"><strong>A.<\/strong> <span class=\"math-tex\">$\\overrightarrow{c}$<\/span> = (1; 2).<\/span><\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2023-05-31 10:10:10","option_type":"math","len":0},{"id":"7009","mon_id":"1158767","chapter_id":"1158890","post_id":"6458","question":"<p>Trong m\u1eb7t ph\u1eb3ng t\u1ecda \u0111\u1ed9 Oxy, cho ba vect\u01a1 <span class=\"math-tex\">$\\overrightarrow{a}$<\/span> = (1; 2), <span class=\"math-tex\">$\\overrightarrow{b}$<\/span> = (4; 3) và <span class=\"math-tex\">$\\overrightarrow{c}$<\/span> = (2; 3).<\/p><p>Tính T = <span class=\"math-tex\">$\\overrightarrow{a}\\bigg(\\overrightarrow{b}+3\\overrightarrow{c}\\bigg)$<\/span> .<\/p>","options":["<strong>A.<\/strong> T = 6.","<strong>B. <\/strong>T = 25.","<strong>C.<\/strong> T = 32.","<strong>D.<\/strong> T = 34."],"correct":"4","level":"2","hint":"","answer":"<p> <span class=\"math-tex\">$\\overrightarrow{c}$<\/span> = (2; 3) ⇒ 3 <span class=\"math-tex\">$\\overrightarrow{c}$<\/span> = (6; 9) ⇒ <span class=\"math-tex\">$\\overrightarrow{b}+3\\overrightarrow{c}$<\/span> = (10; 12).<\/p><p>Suy ra T = <span class=\"math-tex\">$\\overrightarrow{a}\\bigg(\\overrightarrow{b}+3\\overrightarrow{c}\\bigg)$<\/span> = 1.10 + 2.12 = 34.<\/p><p>\u0110áp án \u0111úng là <span style=\"color:#16a085;\"> <strong>D.<\/strong> T = 34.<\/span><\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2023-05-31 10:15:03","option_type":"txt","len":0}]}