{"id":623715,"date":"2026-11-13T09:50:45","date_gmt":"2026-11-13T02:50:45","guid":{"rendered":"https:\/\/hoanghamobile.com\/tin-tuc\/?p=623715"},"modified":"2026-11-13T09:50:45","modified_gmt":"2026-11-13T02:50:45","slug":"cong-thuc-tinh-do-dai-vecto","status":"publish","type":"post","link":"https:\/\/hoanghamobile.com\/tin-tuc\/cong-thuc-tinh-do-dai-vecto\/","title":{"rendered":"C\u00f4ng th\u1ee9c t\u00ednh \u0111\u1ed9 d\u00e0i vect\u01a1 trong Oxy, Oxyz v\u00e0 b\u00e0i t\u1eadp"},"content":{"rendered":"<p><strong>\u0110\u1ed9 d\u00e0i vect\u01a1 l\u00e0 kho\u1ea3ng c\u00e1ch gi\u1eefa \u0111i\u1ec3m \u0111\u1ea7u v\u00e0 \u0111i\u1ec3m cu\u1ed1i c\u1ee7a vect\u01a1, k\u00fd hi\u1ec7u |a\u20d7| ho\u1eb7c |AB\u20d7|. Trong m\u1eb7t ph\u1eb3ng Oxy, vect\u01a1 a\u20d7 = (x; y) c\u00f3 \u0111\u1ed9 d\u00e0i |a\u20d7| = \u221a(x\u00b2 + y\u00b2); vect\u01a1 AB\u20d7 c\u00f3 \u0111\u1ed9 d\u00e0i |AB\u20d7| = \u221a((x<sub>B<\/sub> \u2212 x<sub>A<\/sub>)\u00b2 + (y<sub>B<\/sub> \u2212 y<sub>A<\/sub>)\u00b2).<\/strong> Trong kh\u00f4ng gian Oxyz, ch\u1ec9 c\u1ea7n th\u00eam th\u00e0nh ph\u1ea7n z. D\u01b0\u1edbi \u0111\u00e2y l\u00e0 \u0111\u1ea7y \u0111\u1ee7 c\u00f4ng th\u1ee9c t\u00ednh \u0111\u1ed9 d\u00e0i vect\u01a1, c\u00e1ch t\u00ednh \u0111\u1ed9 d\u00e0i t\u1ed5ng v\u00e0 hi\u1ec7u hai vect\u01a1, k\u00e8m v\u00ed d\u1ee5 c\u00f3 l\u1eddi gi\u1ea3i v\u00e0 b\u00e0i t\u1eadp c\u00f3 \u0111\u00e1p \u00e1n.<\/p>\n<p><strong>T\u00f3m t\u1eaft nhanh:<\/strong><\/p>\n<ul>\n<li><strong>Oxy:<\/strong> |a\u20d7| = \u221a(x\u00b2 + y\u00b2).<\/li>\n<li><strong>Oxyz:<\/strong> |a\u20d7| = \u221a(x\u00b2 + y\u00b2 + z\u00b2).<\/li>\n<li><strong>Vect\u01a1 qua hai \u0111i\u1ec3m:<\/strong> |AB\u20d7| = AB = \u221a((x<sub>B<\/sub> \u2212 x<sub>A<\/sub>)\u00b2 + (y<sub>B<\/sub> \u2212 y<sub>A<\/sub>)\u00b2).<\/li>\n<li><strong>T\u1ed5ng, hi\u1ec7u:<\/strong> |a\u20d7 \u00b1 b\u20d7|\u00b2 = |a\u20d7|\u00b2 + |b\u20d7|\u00b2 \u00b1 2|a\u20d7||b\u20d7|cos(a\u20d7, b\u20d7).<\/li>\n<\/ul>\n<h2>\u0110\u1ed9 d\u00e0i vect\u01a1 l\u00e0 g\u00ec?<\/h2>\n<p>Vect\u01a1 l\u00e0 m\u1ed9t \u0111o\u1ea1n th\u1eb3ng c\u00f3 h\u01b0\u1edbng, g\u1ed3m \u0111i\u1ec3m \u0111\u1ea7u v\u00e0 \u0111i\u1ec3m cu\u1ed1i. \u0110\u1ed9 d\u00e0i c\u1ee7a vect\u01a1 AB\u20d7 ch\u00ednh l\u00e0 \u0111\u1ed9 d\u00e0i \u0111o\u1ea1n th\u1eb3ng AB, k\u00fd hi\u1ec7u |AB\u20d7| = AB. \u0110\u1ed9 d\u00e0i vect\u01a1 lu\u00f4n l\u00e0 m\u1ed9t s\u1ed1 kh\u00f4ng \u00e2m.<\/p>\n<ul>\n<li>Vect\u01a1 c\u00f3 \u0111\u1ed9 d\u00e0i b\u1eb1ng 1 g\u1ecdi l\u00e0 <strong>vect\u01a1 \u0111\u01a1n v\u1ecb<\/strong>.<\/li>\n<li>Vect\u01a1 c\u00f3 \u0111i\u1ec3m \u0111\u1ea7u tr\u00f9ng \u0111i\u1ec3m cu\u1ed1i g\u1ecdi l\u00e0 <strong>vect\u01a1 \u2013 kh\u00f4ng<\/strong>, c\u00f3 \u0111\u1ed9 d\u00e0i b\u1eb1ng 0.<\/li>\n<li>Hai vect\u01a1 b\u1eb1ng nhau khi c\u00f9ng h\u01b0\u1edbng v\u00e0 c\u00f9ng \u0111\u1ed9 d\u00e0i.<\/li>\n<\/ul>\n<p><img fetchpriority=\"high\" decoding=\"async\" class=\"aligncenter size-full wp-image-623716\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/11\/do-dai-vecto-3.jpg\" alt=\"do-dai-vecto-3\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/11\/do-dai-vecto-3.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/11\/do-dai-vecto-3-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/11\/do-dai-vecto-3-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<h2>C\u00f4ng th\u1ee9c t\u00ednh \u0111\u1ed9 d\u00e0i vect\u01a1 trong m\u1eb7t ph\u1eb3ng Oxy<\/h2>\n<h3>Khi bi\u1ebft t\u1ecda \u0111\u1ed9 vect\u01a1<\/h3>\n<p>Cho vect\u01a1 a\u20d7 = (x; y), \u0111\u1ed9 d\u00e0i vect\u01a1 l\u00e0:<\/p>\n<p><strong>|a\u20d7| = \u221a(x\u00b2 + y\u00b2)<\/strong><\/p>\n<p>C\u00f4ng th\u1ee9c n\u00e0y xu\u1ea5t ph\u00e1t t\u1eeb \u0111\u1ecbnh l\u00fd Pytago: x v\u00e0 y l\u00e0 hai c\u1ea1nh g\u00f3c vu\u00f4ng, \u0111\u1ed9 d\u00e0i vect\u01a1 l\u00e0 c\u1ea1nh huy\u1ec1n.<\/p>\n<p><strong>V\u00ed d\u1ee5 1:<\/strong> Cho a\u20d7 = (3; 4). Khi \u0111\u00f3 |a\u20d7| = \u221a(3\u00b2 + 4\u00b2) = \u221a25 = <strong>5<\/strong>.<\/p>\n<h3>Khi bi\u1ebft t\u1ecda \u0111\u1ed9 hai \u0111i\u1ec3m<\/h3>\n<p>Cho A(x<sub>A<\/sub>; y<sub>A<\/sub>) v\u00e0 B(x<sub>B<\/sub>; y<sub>B<\/sub>). Tr\u01b0\u1edbc h\u1ebft t\u00ednh AB\u20d7 = (x<sub>B<\/sub> \u2212 x<sub>A<\/sub>; y<sub>B<\/sub> \u2212 y<sub>A<\/sub>), sau \u0111\u00f3:<\/p>\n<p><strong>|AB\u20d7| = \u221a((x<sub>B<\/sub> \u2212 x<sub>A<\/sub>)\u00b2 + (y<sub>B<\/sub> \u2212 y<sub>A<\/sub>)\u00b2)<\/strong><\/p>\n<p><strong>V\u00ed d\u1ee5 2:<\/strong> Cho A(1; 2), B(4; 6). Ta c\u00f3 AB\u20d7 = (3; 4), n\u00ean |AB\u20d7| = \u221a(9 + 16) = <strong>5<\/strong>.<\/p>\n<p><strong>V\u00ed d\u1ee5 3:<\/strong> Cho M(\u22121; 3), N(2; \u22121). Ta c\u00f3 MN\u20d7 = (3; \u22124), n\u00ean |MN\u20d7| = \u221a(9 + 16) = <strong>5<\/strong>. L\u01b0u \u00fd d\u1ea5u \u00e2m bi\u1ebfn m\u1ea5t khi b\u00ecnh ph\u01b0\u01a1ng.<\/p>\n<h2>C\u00f4ng th\u1ee9c t\u00ednh \u0111\u1ed9 d\u00e0i vect\u01a1 trong kh\u00f4ng gian Oxyz<\/h2>\n<p>Trong kh\u00f4ng gian, vect\u01a1 c\u00f3 th\u00eam th\u00e0nh ph\u1ea7n z. Cho u\u20d7 = (x; y; z):<\/p>\n<p><strong>|u\u20d7| = \u221a(x\u00b2 + y\u00b2 + z\u00b2)<\/strong><\/p>\n<p>V\u1edbi hai \u0111i\u1ec3m A(x<sub>A<\/sub>; y<sub>A<\/sub>; z<sub>A<\/sub>) v\u00e0 B(x<sub>B<\/sub>; y<sub>B<\/sub>; z<sub>B<\/sub>): |AB\u20d7| = \u221a((x<sub>B<\/sub> \u2212 x<sub>A<\/sub>)\u00b2 + (y<sub>B<\/sub> \u2212 y<sub>A<\/sub>)\u00b2 + (z<sub>B<\/sub> \u2212 z<sub>A<\/sub>)\u00b2).<\/p>\n<ul>\n<li><strong>V\u00ed d\u1ee5 4:<\/strong> u\u20d7 = (1; 2; 2) \u21d2 |u\u20d7| = \u221a(1 + 4 + 4) = <strong>3<\/strong>.<\/li>\n<li><strong>V\u00ed d\u1ee5 5:<\/strong> A(1; \u22121; 2), B(3; 1; 3) \u21d2 AB\u20d7 = (2; 2; 1) \u21d2 |AB\u20d7| = \u221a(4 + 4 + 1) = <strong>3<\/strong>.<\/li>\n<\/ul>\n<p>\u0110\u1ed9 d\u00e0i vect\u01a1 c\u0169ng l\u00e0 b\u01b0\u1edbc trung gian khi t\u00ednh kho\u1ea3ng c\u00e1ch, v\u00ed d\u1ee5 trong c\u00f4ng th\u1ee9c <a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/khoang-cach-tu-diem-den-duong-thang\/\">kho\u1ea3ng c\u00e1ch t\u1eeb \u0111i\u1ec3m \u0111\u1ebfn \u0111\u01b0\u1eddng th\u1eb3ng<\/a> ho\u1eb7c khi l\u1eadp <a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/phuong-trinh-duong-thang\/\">ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng<\/a>.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-623717\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/11\/do-dai-vecto-1.jpg\" alt=\"do-dai-vecto-1\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/11\/do-dai-vecto-1.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/11\/do-dai-vecto-1-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/11\/do-dai-vecto-1-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<h2>C\u00e1ch t\u00ednh \u0111\u1ed9 d\u00e0i t\u1ed5ng v\u00e0 hi\u1ec7u hai vect\u01a1<\/h2>\n<h3>Khi bi\u1ebft t\u1ecda \u0111\u1ed9<\/h3>\n<p>T\u00ednh t\u1ecda \u0111\u1ed9 vect\u01a1 t\u1ed5ng (ho\u1eb7c hi\u1ec7u) tr\u01b0\u1edbc, r\u1ed3i \u00e1p d\u1ee5ng c\u00f4ng th\u1ee9c \u0111\u1ed9 d\u00e0i. V\u00ed d\u1ee5 a\u20d7 = (1; 2), b\u20d7 = (2; 2) \u21d2 a\u20d7 + b\u20d7 = (3; 4) \u21d2 |a\u20d7 + b\u20d7| = 5.<\/p>\n<p><strong>V\u00ed d\u1ee5 (Oxyz):<\/strong> Cho a\u20d7 = (1; \u22122; 3), b\u20d7 = (3; 0; 1). T\u00ednh |a\u20d7 + b\u20d7| v\u00e0 |a\u20d7 \u2212 b\u20d7|.<\/p>\n<p><strong>L\u1eddi gi\u1ea3i:<\/strong><\/p>\n<ul>\n<li>a\u20d7 + b\u20d7 = (1 + 3; \u22122 + 0; 3 + 1) = (4; \u22122; 4) \u21d2 |a\u20d7 + b\u20d7| = \u221a(16 + 4 + 16) = \u221a36 = <strong>6<\/strong>.<\/li>\n<li>a\u20d7 \u2212 b\u20d7 = (1 \u2212 3; \u22122 \u2212 0; 3 \u2212 1) = (\u22122; \u22122; 2) \u21d2 |a\u20d7 \u2212 b\u20d7| = \u221a(4 + 4 + 4) = \u221a12 = <strong>2\u221a3<\/strong>.<\/li>\n<\/ul>\n<h3>Khi bi\u1ebft \u0111\u1ed9 d\u00e0i v\u00e0 g\u00f3c gi\u1eefa hai vect\u01a1<\/h3>\n<p>D\u00f9ng t\u00edch v\u00f4 h\u01b0\u1edbng (b\u1ea3n ch\u1ea5t gi\u1ed1ng <a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/dinh-ly-cosin\/\">\u0111\u1ecbnh l\u00fd cosin<\/a> trong tam gi\u00e1c):<\/p>\n<p><strong>|a\u20d7 + b\u20d7|\u00b2 = |a\u20d7|\u00b2 + |b\u20d7|\u00b2 + 2|a\u20d7||b\u20d7|cos(a\u20d7, b\u20d7)<\/strong><\/p>\n<p><strong>|a\u20d7 \u2212 b\u20d7|\u00b2 = |a\u20d7|\u00b2 + |b\u20d7|\u00b2 \u2212 2|a\u20d7||b\u20d7|cos(a\u20d7, b\u20d7)<\/strong><\/p>\n<p><strong>V\u00ed d\u1ee5 6:<\/strong> Cho |a\u20d7| = 3, |b\u20d7| = 4 v\u00e0 g\u00f3c gi\u1eefa hai vect\u01a1 b\u1eb1ng 60\u00b0.<\/p>\n<ul>\n<li>|a\u20d7 + b\u20d7|\u00b2 = 9 + 16 + 2\u00b73\u00b74\u00b7(1\/2) = 37 \u21d2 |a\u20d7 + b\u20d7| = <strong>\u221a37<\/strong>.<\/li>\n<li>|a\u20d7 \u2212 b\u20d7|\u00b2 = 9 + 16 \u2212 12 = 13 \u21d2 |a\u20d7 \u2212 b\u20d7| = <strong>\u221a13<\/strong>.<\/li>\n<\/ul>\n<h3>Trong h\u00ecnh h\u1ecdc (kh\u00f4ng c\u00f3 t\u1ecda \u0111\u1ed9)<\/h3>\n<p>D\u00f9ng quy t\u1eafc <a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/hinh-binh-hanh-la-gi\/\">h\u00ecnh b\u00ecnh h\u00e0nh<\/a> ho\u1eb7c quy t\u1eafc ba \u0111i\u1ec3m \u0111\u1ec3 \u0111\u01b0a t\u1ed5ng vect\u01a1 v\u1ec1 m\u1ed9t vect\u01a1 duy nh\u1ea5t, r\u1ed3i t\u00ednh \u0111\u1ed9 d\u00e0i \u0111o\u1ea1n th\u1eb3ng t\u01b0\u01a1ng \u1ee9ng.<\/p>\n<ul>\n<li><strong>H\u00ecnh vu\u00f4ng ABCD c\u1ea1nh a:<\/strong> AB\u20d7 + AD\u20d7 = AC\u20d7 \u21d2 |AB\u20d7 + AD\u20d7| = AC = <strong>a\u221a2<\/strong>.<\/li>\n<li><strong>Tam gi\u00e1c \u0111\u1ec1u ABC c\u1ea1nh a, M l\u00e0 trung \u0111i\u1ec3m BC:<\/strong> AB\u20d7 + AC\u20d7 = 2AM\u20d7 \u21d2 |AB\u20d7 + AC\u20d7| = 2AM = 2\u00b7(a\u221a3\/2) = <strong>a\u221a3<\/strong>.<\/li>\n<\/ul>\n<p>\u0110\u1ed9 d\u00e0i \u0111\u01b0\u1eddng trung tuy\u1ebfn c\u1ee7a tam gi\u00e1c \u0111\u1ec1u li\u00ean quan \u0111\u1ebfn <a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/trong-tam-la-gi\/\">tr\u1ecdng t\u00e2m tam gi\u00e1c<\/a>, r\u1ea5t hay g\u1eb7p trong b\u00e0i t\u1eadp vect\u01a1.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-623718\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/11\/do-dai-vecto-2.jpg\" alt=\"do-dai-vecto-2\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/11\/do-dai-vecto-2.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/11\/do-dai-vecto-2-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/11\/do-dai-vecto-2-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<h2>Vect\u01a1 \u0111\u01a1n v\u1ecb v\u00e0 c\u00e1ch chu\u1ea9n h\u00f3a vect\u01a1<\/h2>\n<p>Mu\u1ed1n t\u00ecm vect\u01a1 \u0111\u01a1n v\u1ecb c\u00f9ng h\u01b0\u1edbng v\u1edbi a\u20d7 (a\u20d7 \u2260 0\u20d7), ta chia vect\u01a1 cho \u0111\u1ed9 d\u00e0i c\u1ee7a n\u00f3: e\u20d7 = a\u20d7\/|a\u20d7|.<\/p>\n<p>V\u00ed d\u1ee5 a\u20d7 = (3; 4), |a\u20d7| = 5 \u21d2 e\u20d7 = (3\/5; 4\/5). Ki\u1ec3m tra: \u221a(9\/25 + 16\/25) = 1.<\/p>\n<h2>\u1ee8ng d\u1ee5ng \u0111\u1ed9 d\u00e0i vect\u01a1 \u0111\u1ec3 nh\u1eadn d\u1ea1ng tam gi\u00e1c<\/h2>\n<p>\u0110\u1ed9 d\u00e0i vect\u01a1 gi\u00fap ch\u1ee9ng minh tam gi\u00e1c c\u00e2n, \u0111\u1ec1u ho\u1eb7c vu\u00f4ng ch\u1ec9 b\u1eb1ng t\u1ecda \u0111\u1ed9.<\/p>\n<p><strong>V\u00ed d\u1ee5 7:<\/strong> Cho A(1; 1), B(4; 5), C(5; \u22122). Ch\u1ee9ng minh tam gi\u00e1c ABC vu\u00f4ng c\u00e2n t\u1ea1i A.<\/p>\n<ul>\n<li>AB\u20d7 = (3; 4) \u21d2 AB = 5; AC\u20d7 = (4; \u22123) \u21d2 AC = 5. V\u1eady AB = AC, tam gi\u00e1c c\u00e2n t\u1ea1i A.<\/li>\n<li>T\u00edch v\u00f4 h\u01b0\u1edbng AB\u20d7 \u00b7 AC\u20d7 = 3\u00b74 + 4\u00b7(\u22123) = 0 \u21d2 AB \u22a5 AC.<\/li>\n<li>K\u1ebft lu\u1eadn: tam gi\u00e1c ABC <strong>vu\u00f4ng c\u00e2n t\u1ea1i A<\/strong>, v\u00e0 c\u1ea1nh BC = 5\u221a2.<\/li>\n<\/ul>\n<p>Khi \u0111\u00e3 bi\u1ebft \u0111\u1ed9 d\u00e0i c\u00e1c c\u1ea1nh, b\u1ea1n c\u00f3 th\u1ec3 t\u00ednh ti\u1ebfp di\u1ec7n t\u00edch theo c\u00e1c c\u00f4ng th\u1ee9c trong b\u00e0i <a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/dien-tich-tam-giac\/\">di\u1ec7n t\u00edch tam gi\u00e1c<\/a>.<\/p>\n<h2>B\u00e0i t\u1eadp t\u00ednh \u0111\u1ed9 d\u00e0i vect\u01a1 c\u00f3 \u0111\u00e1p \u00e1n<\/h2>\n<table>\n<thead>\n<tr>\n<th>B\u00e0i<\/th>\n<th>\u0110\u1ec1 b\u00e0i<\/th>\n<th>\u0110\u00e1p \u00e1n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1<\/td>\n<td>a\u20d7 = (6; 8). T\u00ednh |a\u20d7|<\/td>\n<td>10<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>A(\u22122; 1), B(1; 5). T\u00ednh |AB\u20d7|<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>u\u20d7 = (2; \u22121; 2). T\u00ednh |u\u20d7|<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>|a\u20d7| = 3, |b\u20d7| = 4, a\u20d7 \u22a5 b\u20d7. T\u00ednh |a\u20d7 + b\u20d7|<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>H\u00ecnh vu\u00f4ng c\u1ea1nh 2. T\u00ednh |AB\u20d7 + AD\u20d7|<\/td>\n<td>2\u221a2<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>a\u20d7 = (3; 4). T\u00ecm vect\u01a1 \u0111\u01a1n v\u1ecb c\u00f9ng h\u01b0\u1edbng<\/td>\n<td>(3\/5; 4\/5)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>L\u1ed7i th\u01b0\u1eddng g\u1eb7p khi t\u00ednh \u0111\u1ed9 d\u00e0i vect\u01a1<\/h2>\n<ul>\n<li><strong>L\u1ea5y t\u1ecda \u0111\u1ed9 \u0111i\u1ec3m \u0111\u1ea7u tr\u1eeb \u0111i\u1ec3m cu\u1ed1i:<\/strong> AB\u20d7 = (x<sub>B<\/sub> \u2212 x<sub>A<\/sub>; \u2026), nh\u01b0ng v\u1edbi \u0111\u1ed9 d\u00e0i th\u00ec th\u1ee9 t\u1ef1 kh\u00f4ng \u1ea3nh h\u01b0\u1edfng v\u00ec \u0111\u00e3 b\u00ecnh ph\u01b0\u01a1ng.<\/li>\n<li><strong>Qu\u00ean c\u0103n b\u1eadc hai:<\/strong> x\u00b2 + y\u00b2 l\u00e0 b\u00ecnh ph\u01b0\u01a1ng \u0111\u1ed9 d\u00e0i, kh\u00f4ng ph\u1ea3i \u0111\u1ed9 d\u00e0i.<\/li>\n<li><strong>C\u1ed9ng \u0111\u1ed9 d\u00e0i thay v\u00ec c\u1ed9ng vect\u01a1:<\/strong> |a\u20d7 + b\u20d7| n\u00f3i chung kh\u00f4ng b\u1eb1ng |a\u20d7| + |b\u20d7|; ch\u1ec9 b\u1eb1ng khi hai vect\u01a1 c\u00f9ng h\u01b0\u1edbng.<\/li>\n<li><strong>Sai d\u1ea5u cos:<\/strong> g\u00f3c t\u00f9 c\u00f3 cos \u00e2m, c\u1ea7n gi\u1eef nguy\u00ean d\u1ea5u khi thay v\u00e0o c\u00f4ng th\u1ee9c.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-623719\" src=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/11\/do-dai-vecto-4.jpg\" alt=\"do-dai-vecto-4\" width=\"800\" height=\"450\" srcset=\"https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/11\/do-dai-vecto-4.jpg 800w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/11\/do-dai-vecto-4-300x169.jpg 300w, https:\/\/hoanghamobile.com\/tin-tuc\/wp-content\/uploads\/2026\/11\/do-dai-vecto-4-768x432.jpg 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<h2>C\u00e2u h\u1ecfi th\u01b0\u1eddng g\u1eb7p v\u1ec1 \u0111\u1ed9 d\u00e0i vect\u01a1<\/h2>\n<h3>C\u00f4ng th\u1ee9c t\u00ednh \u0111\u1ed9 d\u00e0i vect\u01a1 l\u00e0 g\u00ec?<\/h3>\n<p>V\u1edbi vect\u01a1 a\u20d7 = (x; y) trong m\u1eb7t ph\u1eb3ng, |a\u20d7| = \u221a(x\u00b2 + y\u00b2). Trong kh\u00f4ng gian, a\u20d7 = (x; y; z) th\u00ec |a\u20d7| = \u221a(x\u00b2 + y\u00b2 + z\u00b2).<\/p>\n<h3>\u0110\u1ed9 d\u00e0i vect\u01a1 AB b\u1eb1ng g\u00ec?<\/h3>\n<p>\u0110\u1ed9 d\u00e0i vect\u01a1 AB\u20d7 b\u1eb1ng \u0111\u1ed9 d\u00e0i \u0111o\u1ea1n th\u1eb3ng AB, t\u00ednh b\u1eb1ng \u221a((x<sub>B<\/sub> \u2212 x<sub>A<\/sub>)\u00b2 + (y<sub>B<\/sub> \u2212 y<sub>A<\/sub>)\u00b2) trong m\u1eb7t ph\u1eb3ng t\u1ecda \u0111\u1ed9.<\/p>\n<h3>\u0110\u1ed9 d\u00e0i vect\u01a1 c\u00f3 th\u1ec3 \u00e2m kh\u00f4ng?<\/h3>\n<p>Kh\u00f4ng. \u0110\u1ed9 d\u00e0i vect\u01a1 lu\u00f4n l\u1edbn h\u01a1n ho\u1eb7c b\u1eb1ng 0; b\u1eb1ng 0 khi v\u00e0 ch\u1ec9 khi \u0111\u00f3 l\u00e0 vect\u01a1 \u2013 kh\u00f4ng.<\/p>\n<h3>Vect\u01a1 \u0111\u01a1n v\u1ecb l\u00e0 g\u00ec?<\/h3>\n<p>Vect\u01a1 \u0111\u01a1n v\u1ecb l\u00e0 vect\u01a1 c\u00f3 \u0111\u1ed9 d\u00e0i b\u1eb1ng 1. Mu\u1ed1n c\u00f3 vect\u01a1 \u0111\u01a1n v\u1ecb c\u00f9ng h\u01b0\u1edbng v\u1edbi a\u20d7, ta l\u1ea5y a\u20d7 chia cho |a\u20d7|.<\/p>\n<h3>L\u00e0m sao t\u00ednh \u0111\u1ed9 d\u00e0i t\u1ed5ng hai vect\u01a1 khi bi\u1ebft g\u00f3c?<\/h3>\n<p>D\u00f9ng c\u00f4ng th\u1ee9c |a\u20d7 + b\u20d7|\u00b2 = |a\u20d7|\u00b2 + |b\u20d7|\u00b2 + 2|a\u20d7||b\u20d7|cos(a\u20d7, b\u20d7), r\u1ed3i l\u1ea5y c\u0103n b\u1eadc hai k\u1ebft qu\u1ea3.<\/p>\n<h3>\u0110\u1ed9 d\u00e0i vect\u01a1 h\u1ecdc \u1edf l\u1edbp m\u1ea5y?<\/h3>\n<p>Kh\u00e1i ni\u1ec7m \u0111\u1ed9 d\u00e0i vect\u01a1 v\u00e0 t\u1ecda \u0111\u1ed9 vect\u01a1 trong m\u1eb7t ph\u1eb3ng \u0111\u01b0\u1ee3c h\u1ecdc \u1edf To\u00e1n l\u1edbp 10; vect\u01a1 trong kh\u00f4ng gian Oxyz \u0111\u01b0\u1ee3c h\u1ecdc \u1edf To\u00e1n l\u1edbp 12.<\/p>\n<h2>K\u1ebft lu\u1eadn<\/h2>\n<p>\u0110\u1ed9 d\u00e0i vect\u01a1 l\u00e0 \u0111\u1ed9 d\u00e0i \u0111o\u1ea1n th\u1eb3ng n\u1ed1i \u0111i\u1ec3m \u0111\u1ea7u v\u00e0 \u0111i\u1ec3m cu\u1ed1i, t\u00ednh b\u1eb1ng \u221a(x\u00b2 + y\u00b2) trong m\u1eb7t ph\u1eb3ng v\u00e0 \u221a(x\u00b2 + y\u00b2 + z\u00b2) trong kh\u00f4ng gian. Khi g\u1eb7p t\u1ed5ng ho\u1eb7c hi\u1ec7u vect\u01a1, h\u00e3y t\u00ednh t\u1ecda \u0111\u1ed9 vect\u01a1 m\u1edbi ho\u1eb7c d\u00f9ng c\u00f4ng th\u1ee9c t\u00edch v\u00f4 h\u01b0\u1edbng v\u1edbi g\u00f3c gi\u1eefa hai vect\u01a1. Luy\u1ec7n b\u1ea3ng b\u00e0i t\u1eadp tr\u00ean s\u1ebd gi\u00fap b\u1ea1n l\u00e0m nhanh d\u1ea1ng to\u00e1n n\u00e0y trong \u0111\u1ec1 ki\u1ec3m tra.<\/p>\n<p>Xem th\u00eam:<\/p>\n<ul>\n<li><a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/vecto-phap-tuyen\/\">Vect\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 g\u00ec? C\u00e1ch t\u00ecm v\u00e0 b\u00e0i t\u1eadp c\u00f3 l\u1eddi gi\u1ea3i<\/a><\/li>\n<li><a href=\"https:\/\/hoanghamobile.com\/tin-tuc\/phuong-trinh-mat-phang\/\">Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng: c\u00f4ng th\u1ee9c, c\u00e1ch vi\u1ebft v\u00e0 v\u00ed d\u1ee5<\/a><\/li>\n<\/ul>\n<p><script type=\"application\/ld+json\">{\"@context\": \"https:\/\/schema.org\", \"@type\": \"FAQPage\", \"mainEntity\": [{\"@type\": \"Question\", \"name\": \"C\u00f4ng th\u1ee9c t\u00ednh \u0111\u1ed9 d\u00e0i vect\u01a1 l\u00e0 g\u00ec?\", \"acceptedAnswer\": {\"@type\": \"Answer\", \"text\": \"V\u1edbi vect\u01a1 a\u20d7 = (x; y) trong m\u1eb7t ph\u1eb3ng, |a\u20d7| = \u221a(x\u00b2 + y\u00b2). Trong kh\u00f4ng gian, a\u20d7 = (x; y; z) th\u00ec |a\u20d7| = \u221a(x\u00b2 + y\u00b2 + z\u00b2).\"}}, {\"@type\": \"Question\", \"name\": \"\u0110\u1ed9 d\u00e0i vect\u01a1 AB b\u1eb1ng g\u00ec?\", \"acceptedAnswer\": {\"@type\": \"Answer\", \"text\": \"\u0110\u1ed9 d\u00e0i vect\u01a1 AB\u20d7 b\u1eb1ng \u0111\u1ed9 d\u00e0i \u0111o\u1ea1n th\u1eb3ng AB, t\u00ednh b\u1eb1ng \u221a((x_B \u2212 xA)\u00b2 + (yB \u2212 yA)\u00b2) trong m\u1eb7t ph\u1eb3ng t\u1ecda \u0111\u1ed9.\"}}, {\"@type\": \"Question\", \"name\": \"\u0110\u1ed9 d\u00e0i vect\u01a1 c\u00f3 th\u1ec3 \u00e2m kh\u00f4ng?\", \"acceptedAnswer\": {\"@type\": \"Answer\", \"text\": \"Kh\u00f4ng. \u0110\u1ed9 d\u00e0i vect\u01a1 lu\u00f4n l\u1edbn h\u01a1n ho\u1eb7c b\u1eb1ng 0; b\u1eb1ng 0 khi v\u00e0 ch\u1ec9 khi \u0111\u00f3 l\u00e0 vect\u01a1 \u2013 kh\u00f4ng.\"}}, {\"@type\": \"Question\", \"name\": \"Vect\u01a1 \u0111\u01a1n v\u1ecb l\u00e0 g\u00ec?\", \"acceptedAnswer\": {\"@type\": \"Answer\", \"text\": \"Vect\u01a1 \u0111\u01a1n v\u1ecb l\u00e0 vect\u01a1 c\u00f3 \u0111\u1ed9 d\u00e0i b\u1eb1ng 1. Mu\u1ed1n c\u00f3 vect\u01a1 \u0111\u01a1n v\u1ecb c\u00f9ng h\u01b0\u1edbng v\u1edbi a\u20d7, ta l\u1ea5y a\u20d7 chia cho |a\u20d7|.\"}}, {\"@type\": \"Question\", \"name\": \"L\u00e0m sao t\u00ednh \u0111\u1ed9 d\u00e0i t\u1ed5ng hai vect\u01a1 khi bi\u1ebft g\u00f3c?\", \"acceptedAnswer\": {\"@type\": \"Answer\", \"text\": \"D\u00f9ng c\u00f4ng th\u1ee9c |a\u20d7 + b\u20d7|\u00b2 = |a\u20d7|\u00b2 + |b\u20d7|\u00b2 + 2|a\u20d7||b\u20d7|cos(a\u20d7, b\u20d7), r\u1ed3i l\u1ea5y c\u0103n b\u1eadc hai k\u1ebft qu\u1ea3.\"}}, {\"@type\": \"Question\", \"name\": \"\u0110\u1ed9 d\u00e0i vect\u01a1 h\u1ecdc \u1edf l\u1edbp m\u1ea5y?\", \"acceptedAnswer\": {\"@type\": \"Answer\", \"text\": \"Kh\u00e1i ni\u1ec7m \u0111\u1ed9 d\u00e0i vect\u01a1 v\u00e0 t\u1ecda \u0111\u1ed9 vect\u01a1 trong m\u1eb7t ph\u1eb3ng \u0111\u01b0\u1ee3c h\u1ecdc \u1edf To\u00e1n l\u1edbp 10; vect\u01a1 trong kh\u00f4ng gian Oxyz \u0111\u01b0\u1ee3c h\u1ecdc \u1edf To\u00e1n l\u1edbp 12.\"}}]}<\/script><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0110\u1ed9 d\u00e0i vect\u01a1 l\u00e0 kho\u1ea3ng c\u00e1ch gi\u1eefa \u0111i\u1ec3m \u0111\u1ea7u v\u00e0 \u0111i\u1ec3m cu\u1ed1i c\u1ee7a vect\u01a1, k\u00fd hi\u1ec7u |a\u20d7| ho\u1eb7c |AB\u20d7|. Trong m\u1eb7t ph\u1eb3ng Oxy, vect\u01a1 a\u20d7 = (x; y) c\u00f3 \u0111\u1ed9 d\u00e0i |a\u20d7| = \u221a(x\u00b2 + y\u00b2); vect\u01a1 AB\u20d7 c\u00f3 \u0111\u1ed9 d\u00e0i |AB\u20d7| = \u221a((xB \u2212 xA)\u00b2 + (yB \u2212 yA)\u00b2). Trong kh\u00f4ng gian Oxyz, ch\u1ec9 [&hellip;]<\/p>\n","protected":false},"author":102,"featured_media":623720,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"c\u00f4ng th\u1ee9c t\u00ednh \u0111\u1ed9 d\u00e0i vect\u01a1","_yoast_wpseo_title":"C\u00f4ng th\u1ee9c t\u00ednh \u0111\u1ed9 d\u00e0i vect\u01a1 trong Oxy, Oxyz v\u00e0 b\u00e0i t\u1eadp","_yoast_wpseo_metadesc":"C\u00f4ng th\u1ee9c t\u00ednh \u0111\u1ed9 d\u00e0i vect\u01a1 trong Oxy, Oxyz, \u0111\u1ed9 d\u00e0i vect\u01a1 AB, t\u1ed5ng v\u00e0 hi\u1ec7u hai vect\u01a1 k\u00e8m v\u00ed d\u1ee5 c\u00f3 l\u1eddi gi\u1ea3i. Xem chi ti\u1ebft t\u1ea1i Ho\u00e0ng H\u00e0 Mobile.","footnotes":""},"categories":[3910,30453],"tags":[],"class_list":["post-623715","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-kham-pha","category-kien-thuc"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v27.3 (Yoast SEO v27.4) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>C\u00f4ng th\u1ee9c t\u00ednh \u0111\u1ed9 d\u00e0i vect\u01a1 trong Oxy, Oxyz v\u00e0 b\u00e0i t\u1eadp<\/title>\n<meta name=\"description\" content=\"C\u00f4ng th\u1ee9c t\u00ednh \u0111\u1ed9 d\u00e0i vect\u01a1 trong Oxy, Oxyz, \u0111\u1ed9 d\u00e0i vect\u01a1 AB, t\u1ed5ng v\u00e0 hi\u1ec7u hai vect\u01a1 k\u00e8m v\u00ed d\u1ee5 c\u00f3 l\u1eddi gi\u1ea3i. 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