{"id":2283,"date":"2026-04-25T02:39:02","date_gmt":"2026-04-25T02:39:02","guid":{"rendered":"https:\/\/77.buffdemo.com\/index.php\/2026\/04\/25\/toan-12-chuong-2-vector-va-he-truc-toa-do\/"},"modified":"2026-04-25T02:39:02","modified_gmt":"2026-04-25T02:39:02","slug":"toan-12-chuong-2-vector-va-he-truc-toa-do","status":"publish","type":"post","link":"https:\/\/77.buffdemo.com\/index.php\/2026\/04\/25\/toan-12-chuong-2-vector-va-he-truc-toa-do\/","title":{"rendered":"To\u00e1n 12 Ch\u01b0\u01a1ng 2: Vector v\u00e0 h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 kh\u00f4ng gian"},"content":{"rendered":"<div class=\"entry-content single-page\">\n<p>Chia s\u1ebb t\u00e0i li\u1ec7u To\u00e1n 12 Ch\u01b0\u01a1ng 2: Vector v\u00e0 h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 kh\u00f4ng gian c\u00f3 l\u1eddi gi\u1ea3i chi ti\u1ebft, b\u00e1m s\u00e1t ch\u01b0\u01a1ng tr\u00ecnh m\u1edbi.<\/p>\n<div class=\"code-block code-block-2\" style=\"margin: 8px auto; text-align: center; display: block; clear: both;\">\n<div> <\/div>\n<\/div>\n<h2 class=\"wp-block-heading has-vivid-red-color has-text-color has-link-color wp-elements-c213465a9013b9ae3517e0e3a5e7ed42\" id=\"h-n\u1ed9i-dung-tai-li\u1ec7u-ch\u01b0\u01a1ng-vector-va-h\u1ec7-tr\u1ee5c-t\u1ecda-d\u1ed9-khong-gian\"><span class=\"ez-toc-section\" id=\"Noi_dung_tai_lieu_chuong_Vector_va_he_truc_toa_do_khong_gian\"><\/span>N\u1ed9i dung t\u00e0i li\u1ec7u ch\u01b0\u01a1ng Vector v\u00e0 h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 kh\u00f4ng gian<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u0110\u00e2y l\u00e0 m\u1ed9t b\u1ed9 t\u00e0i li\u1ec7u h\u1ecdc t\u1eadp to\u00e0n di\u1ec7n cho Ch\u01b0\u01a1ng II (Vect\u01a1 v\u00e0 H\u1ec7 t\u1ecda \u0111\u1ed9 trong kh\u00f4ng gian) m\u00f4n To\u00e1n 12 theo ch\u01b0\u01a1ng tr\u00ecnh m\u1edbi.<\/p>\n<h3 class=\"wp-block-heading\" id=\"h-cac-chuyen-d\u1ec1-ch\u01b0\u01a1ng-vector-l\u1edbp-12\"><span class=\"ez-toc-section\" id=\"Cac_chuyen_de_chuong_Vector_lop_12\"><\/span><strong>C\u00e1c chuy\u00ean \u0111\u1ec1 ch\u01b0\u01a1ng Vector l\u1edbp 12<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<figure class=\"wp-block-table\">\n<table>\n<thead>\n<tr>\n<th>Ch\u1ee7 \u0111\u1ec1<\/th>\n<th>T\u00f3m t\u1eaft n\u1ed9i dung<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Vect\u01a1 v\u00e0 c\u00e1c ph\u00e9p to\u00e1n trong kh\u00f4ng gian<\/td>\n<td>Tr\u00ecnh b\u00e0y \u0111\u1ecbnh ngh\u0129a vect\u01a1 trong kh\u00f4ng gian, c\u00e1c kh\u00e1i ni\u1ec7m li\u00ean quan (gi\u00e1, \u0111\u1ed9 d\u00e0i, c\u00f9ng ph\u01b0\u01a1ng\/h\u01b0\u1edbng), t\u1ed5ng v\u00e0 hi\u1ec7u c\u1ee7a hai vect\u01a1 (quy t\u1eafc ba \u0111i\u1ec3m, quy t\u1eafc h\u00ecnh b\u00ecnh h\u00e0nh), quy t\u1eafc h\u00ecnh h\u1ed9p, v\u00e0 t\u00edch c\u1ee7a m\u1ed9t s\u1ed1 v\u1edbi m\u1ed9t vect\u01a1. <br \/>Bao g\u1ed3m c\u00e1c v\u00ed d\u1ee5 minh h\u1ecda v\u00e0 l\u1eddi gi\u1ea3i chi ti\u1ebft<\/td>\n<\/tr>\n<tr>\n<td>T\u1ecda \u0111\u1ed9 c\u1ee7a vect\u01a1 trong kh\u00f4ng gian<\/td>\n<td>Gi\u1edbi thi\u1ec7u v\u1ec1 h\u1ec7 t\u1ecda \u0111\u1ed9 Descartes vu\u00f4ng g\u00f3c <img loading=\"lazy\" decoding=\"async\" alt=\"Oxyz\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"16\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-b3ba791f71f309ec29e53aa23ad262c0_l3.png\" style=\"vertical-align: -4px;\" title=\"Rendered by QuickLaTeX.com\" width=\"42\"\/>, t\u1ecda \u0111\u1ed9 c\u1ee7a \u0111i\u1ec3m v\u00e0 vect\u01a1, vect\u01a1 \u0111\u01a1n v\u1ecb <img loading=\"lazy\" decoding=\"async\" alt=\"\\vec{i}, \\vec{j}, \\vec{k}\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"22\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-e10165b41e55b5d104d644468b04b0e6_l3.png\" style=\"vertical-align: -4px;\" title=\"Rendered by QuickLaTeX.com\" width=\"41\"\/>, v\u00e0 \u0111i\u1ec1u ki\u1ec7n hai vect\u01a1 b\u1eb1ng nhau. <br \/>Bao g\u1ed3m c\u00e1c d\u1ea1ng to\u00e1n x\u00e1c \u0111\u1ecbnh t\u1ecda \u0111\u1ed9 \u0111i\u1ec3m\/vect\u01a1<\/td>\n<\/tr>\n<tr>\n<td>Bi\u1ec3u th\u1ee9c t\u1ecda \u0111\u1ed9 c\u1ee7a c\u00e1c ph\u00e9p to\u00e1n vect\u01a1<\/td>\n<td>Cung c\u1ea5p c\u00f4ng th\u1ee9c t\u1ecda \u0111\u1ed9 cho t\u1ed5ng\/hi\u1ec7u hai vect\u01a1, t\u00edch c\u1ee7a m\u1ed9t s\u1ed1 v\u1edbi m\u1ed9t vect\u01a1, v\u00e0 t\u00edch v\u00f4 h\u01b0\u1edbng c\u1ee7a hai vect\u01a1. <br \/>N\u00eau c\u00e1c \u1ee9ng d\u1ee5ng nh\u01b0 t\u00ednh \u0111\u1ed9 d\u00e0i \u0111o\u1ea1n th\u1eb3ng, x\u00e1c \u0111\u1ecbnh t\u1ecda \u0111\u1ed9 trung \u0111i\u1ec3m, tr\u1ecdng t\u00e2m tam gi\u00e1c, v\u00e0 t\u00ednh g\u00f3c gi\u1eefa hai vect\u01a1<\/td>\n<\/tr>\n<tr>\n<td>\u0110\u1ec1 \u00f4n t\u1eadp cu\u1ed1i ch\u01b0\u01a1ng<\/td>\n<td>Bao g\u1ed3m 3 \u0111\u1ec1 \u00f4n t\u1eadp t\u1ed5ng h\u1ee3p (\u0110\u1ec1 s\u1ed1 01, \u0110\u1ec1 s\u1ed1 02, \u0110\u1ec1 s\u1ed1 03) d\u01b0\u1edbi d\u1ea1ng tr\u1eafc nghi\u1ec7m nhi\u1ec1u ph\u01b0\u01a1ng \u00e1n, tr\u1eafc nghi\u1ec7m \u0111\u00fang\/sai v\u00e0 c\u00e2u tr\u1ea3 l\u1eddi ng\u1eafn\/t\u1ef1 lu\u1eadn. <br \/>C\u00e1c \u0111\u1ec1 n\u00e0y ki\u1ec3m tra to\u00e0n b\u1ed9 ki\u1ebfn th\u1ee9c v\u1ec1 vect\u01a1 v\u00e0 t\u1ecda \u0111\u1ed9 trong kh\u00f4ng gian<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/figure>\n<p class=\"has-text-align-left\"><strong>Xem tr\u01b0\u1edbc t\u00e0i li\u1ec7u:<\/strong><\/p>\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" alt=\"To\u00e1n 12 Vector v\u00e0 h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 kh\u00f4ng gian\" class=\"wp-image-8442\" height=\"675\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/image-16-1.jpg\" width=\"1178\"\/><\/figure>\n<h3 class=\"wp-block-heading\" id=\"h-cac-d\u1ea1ng-bai-t\u1eadp-ch\u01b0\u01a1ng-vector-12\"><span class=\"ez-toc-section\" id=\"Cac_dang_bai_tap_chuong_Vector_12\"><\/span>C\u00e1c d\u1ea1ng b\u00e0i t\u1eadp ch\u01b0\u01a1ng Vector 12<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>I. D\u1ea1ng b\u00e0i t\u1eadp v\u1ec1 Vect\u01a1 v\u00e0 c\u00e1c Ph\u00e9p to\u00e1n (H\u00ecnh h\u1ecdc thu\u1ea7n t\u00fay)<\/strong><\/p>\n<p>C\u00e1c b\u00e0i t\u1eadp n\u00e0y kh\u00f4ng y\u00eau c\u1ea7u s\u1eed d\u1ee5ng h\u1ec7 t\u1ecda \u0111\u1ed9 <img loading=\"lazy\" decoding=\"async\" alt=\"Oxyz\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"16\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-b3ba791f71f309ec29e53aa23ad262c0_l3.png\" style=\"vertical-align: -4px;\" title=\"Rendered by QuickLaTeX.com\" width=\"42\"\/>, m\u00e0 d\u1ef1a tr\u00ean c\u00e1c quy t\u1eafc h\u00ecnh h\u1ecdc:<\/p>\n<ul class=\"wp-block-list\">\n<li><strong>X\u00e1c \u0111\u1ecbnh v\u00e0 Ph\u00e2n t\u00edch Vect\u01a1:<\/strong>\n<ul class=\"wp-block-list\">\n<li>T\u00ecm c\u00e1c vect\u01a1 b\u1eb1ng nhau, vect\u01a1 \u0111\u1ed1i nhau.<\/li>\n<li>\u00c1p d\u1ee5ng\u00a0Quy t\u1eafc 3 \u0111i\u1ec3m, Quy t\u1eafc h\u00ecnh b\u00ecnh h\u00e0nh\u00a0\u0111\u1ec3 t\u00ednh t\u1ed5ng\/hi\u1ec7u c\u00e1c vect\u01a1.<\/li>\n<li>\u00c1p d\u1ee5ng\u00a0Quy t\u1eafc h\u00ecnh h\u1ed9p\u00a0\u0111\u1ec3 t\u00ednh t\u1ed5ng <img loading=\"lazy\" decoding=\"async\" alt=\"\\vec{AB} + \\vec{AD} + \\vec{AA'} = \\vec{AC'}\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"20\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-df0cedb0bc15f3232423694b2c4fe88d_l3.png\" style=\"vertical-align: -2px;\" title=\"Rendered by QuickLaTeX.com\" width=\"185\"\/>.<\/li>\n<li>Ph\u00e2n t\u00edch m\u1ed9t vect\u01a1\u00a0theo ba vect\u01a1 kh\u00f4ng \u0111\u1ed3ng ph\u1eb3ng cho tr\u01b0\u1edbc.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Ch\u1ee9ng minh \u0110\u1eb3ng th\u1ee9c Vect\u01a1:<\/strong>\n<ul class=\"wp-block-list\">\n<li>S\u1eed d\u1ee5ng c\u00e1c quy t\u1eafc c\u1ed9ng, tr\u1eeb v\u00e0 nh\u00e2n m\u1ed9t s\u1ed1 v\u1edbi vect\u01a1 \u0111\u1ec3 ch\u1ee9ng minh c\u00e1c m\u1ed1i li\u00ean h\u1ec7 gi\u1eefa c\u00e1c vect\u01a1 trong c\u00e1c h\u00ecnh kh\u00f4ng gian (t\u1ee9 di\u1ec7n, h\u00ecnh h\u1ed9p, l\u0103ng tr\u1ee5).<\/li>\n<\/ul>\n<\/li>\n<li><strong>B\u00e0i to\u00e1n th\u1ef1c t\u1ebf v\u1ec1 L\u1ef1c:<\/strong>\n<ul class=\"wp-block-list\">\n<li>T\u00ednh \u0111\u1ed9 l\u1edbn h\u1ee3p l\u1ef1c c\u1ee7a nhi\u1ec1u l\u1ef1c t\u00e1c \u0111\u1ed9ng v\u00e0o m\u1ed9t v\u1eadt (d\u1ef1a tr\u00ean quy t\u1eafc c\u1ed9ng vect\u01a1 v\u00e0 \u0111\u1ecbnh l\u00fd cosin).<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>II. D\u1ea1ng b\u00e0i t\u1eadp v\u1ec1 T\u1ecda \u0111\u1ed9 \u0110i\u1ec3m v\u00e0 Vect\u01a1<\/strong><\/p>\n<p>C\u00e1c b\u00e0i t\u1eadp n\u00e0y li\u00ean quan \u0111\u1ebfn vi\u1ec7c x\u00e1c \u0111\u1ecbnh v\u1ecb tr\u00ed c\u1ee7a c\u00e1c \u0111\u1ed1i t\u01b0\u1ee3ng trong h\u1ec7 t\u1ecda \u0111\u1ed9 <img loading=\"lazy\" decoding=\"async\" alt=\"Oxyz\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"16\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-b3ba791f71f309ec29e53aa23ad262c0_l3.png\" style=\"vertical-align: -4px;\" title=\"Rendered by QuickLaTeX.com\" width=\"42\"\/>:<\/p>\n<ul class=\"wp-block-list\">\n<li><strong>X\u00e1c \u0111\u1ecbnh T\u1ecda \u0111\u1ed9 c\u01a1 b\u1ea3n:<\/strong>\n<ul class=\"wp-block-list\">\n<li>T\u00ecm t\u1ecda \u0111\u1ed9 c\u1ee7a vect\u01a1 <img loading=\"lazy\" decoding=\"async\" alt=\"\\vec{AB}\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"17\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-2648d55b95b5b8961411bbc47fd69d9d_l3.png\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" width=\"27\"\/> khi bi\u1ebft t\u1ecda \u0111\u1ed9 \u0111i\u1ec3m \u0111\u1ea7u <img loading=\"lazy\" decoding=\"async\" alt=\"A\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"13\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-531edf482672504c54f73b283e015b15_l3.png\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" width=\"13\"\/> v\u00e0 \u0111i\u1ec3m cu\u1ed1i <img loading=\"lazy\" decoding=\"async\" alt=\"B\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"12\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-08ca5abc811abdf360a2a774095d997e_l3.png\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" width=\"14\"\/>.<\/li>\n<li>T\u00ecm t\u1ecda \u0111\u1ed9 c\u1ee7a m\u1ed9t \u0111i\u1ec3m <img loading=\"lazy\" decoding=\"async\" alt=\"N\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"12\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-d26e9f72d1ca8188a9f52e0867ccf785_l3-1.png\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" width=\"16\"\/> khi bi\u1ebft t\u1ecda \u0111\u1ed9 \u0111i\u1ec3m <img loading=\"lazy\" decoding=\"async\" alt=\"M\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"12\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-279c67ecc79df39a3fb1845eddf29352_l3-1.png\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" width=\"19\"\/> v\u00e0 vect\u01a1 <img loading=\"lazy\" decoding=\"async\" alt=\"\\vec{MN}\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"17\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-a41a356792bdb1938aec618c43694e0d_l3.png\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" width=\"35\"\/>.<\/li>\n<li>T\u00ecm t\u1ecda \u0111\u1ed9 c\u1ee7a \u0111i\u1ec3m <img loading=\"lazy\" decoding=\"async\" alt=\"A\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"13\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-531edf482672504c54f73b283e015b15_l3.png\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" width=\"13\"\/>, <img loading=\"lazy\" decoding=\"async\" alt=\"B\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"12\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-08ca5abc811abdf360a2a774095d997e_l3.png\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" width=\"14\"\/>, <img loading=\"lazy\" decoding=\"async\" alt=\"C\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"12\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-db7f9cfea6592e6b2471b6ec08344c50_l3.png\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" width=\"14\"\/> khi bi\u1ebft vect\u01a1 <img loading=\"lazy\" decoding=\"async\" alt=\"\\vec{OA}\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"17\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-e9c059409e9c2d0d039725da729c364c_l3.png\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" width=\"27\"\/>, <img loading=\"lazy\" decoding=\"async\" alt=\"\\vec{OB}\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"17\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-5b728edb3adb029d3c216ecc8698fd26_l3.png\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" width=\"28\"\/>, <img loading=\"lazy\" decoding=\"async\" alt=\"\\vec{OC}\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"17\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-95b7121ea328c37d576ebdef122ca7b7_l3.png\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" width=\"28\"\/> \u0111\u01b0\u1ee3c bi\u1ec3u di\u1ec5n qua c\u00e1c vect\u01a1 \u0111\u01a1n v\u1ecb <img loading=\"lazy\" decoding=\"async\" alt=\"\\vec{i}, \\vec{j}, \\vec{k}\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"22\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-e10165b41e55b5d104d644468b04b0e6_l3.png\" style=\"vertical-align: -4px;\" title=\"Rendered by QuickLaTeX.com\" width=\"41\"\/>.<\/li>\n<\/ul>\n<\/li>\n<li><strong>T\u1ecda \u0111\u1ed9 \u0110i\u1ec3m \u0111\u1eb7c bi\u1ec7t:<\/strong>\n<ul class=\"wp-block-list\">\n<li>T\u00ecm t\u1ecda \u0111\u1ed9\u00a0Trung \u0111i\u1ec3m\u00a0c\u1ee7a \u0111o\u1ea1n th\u1eb3ng.<\/li>\n<li>T\u00ecm t\u1ecda \u0111\u1ed9\u00a0Tr\u1ecdng t\u00e2m\u00a0<img loading=\"lazy\" decoding=\"async\" alt=\"G\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"12\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-b0a928fb22de85ee5c21075ca6521836_l3.png\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" width=\"14\"\/> c\u1ee7a tam gi\u00e1c ho\u1eb7c t\u1ee9 di\u1ec7n.<\/li>\n<li>T\u00ecm t\u1ecda \u0111\u1ed9\u00a0H\u00ecnh chi\u1ebfu vu\u00f4ng g\u00f3c\u00a0c\u1ee7a m\u1ed9t \u0111i\u1ec3m l\u00ean c\u00e1c tr\u1ee5c t\u1ecda \u0111\u1ed9 (<img loading=\"lazy\" decoding=\"async\" alt=\"Ox, Oy, Oz\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"16\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-010582761d9261606550567d144611cc_l3.png\" style=\"vertical-align: -4px;\" title=\"Rendered by QuickLaTeX.com\" width=\"86\"\/>) ho\u1eb7c c\u00e1c m\u1eb7t ph\u1eb3ng t\u1ecda \u0111\u1ed9 (<img loading=\"lazy\" decoding=\"async\" alt=\"Oxy, Oyz, Oxz\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"16\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-20e27ff15d08af4079368842e2750c42_l3.png\" style=\"vertical-align: -4px;\" title=\"Rendered by QuickLaTeX.com\" width=\"115\"\/>).<\/li>\n<li>T\u00ecm t\u1ecda \u0111\u1ed9\u00a0\u0110i\u1ec3m \u0111\u1ed1i x\u1ee9ng\u00a0c\u1ee7a m\u1ed9t \u0111i\u1ec3m qua m\u1ed9t \u0111i\u1ec3m, tr\u1ee5c ho\u1eb7c m\u1eb7t ph\u1eb3ng.<\/li>\n<\/ul>\n<\/li>\n<li><strong>\u0110i\u1ec1u ki\u1ec7n v\u1ec1 \u0110i\u1ec3m:<\/strong>\n<ul class=\"wp-block-list\">\n<li>T\u00ecm t\u1ecda \u0111\u1ed9 \u0111i\u1ec3m th\u1ecfa m\u00e3n \u0111i\u1ec1u ki\u1ec7n th\u1eb3ng h\u00e0ng (v\u00ed d\u1ee5: <img loading=\"lazy\" decoding=\"async\" alt=\"A, B, M\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"17\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-3fde417297bcd5546f287e534a055caf_l3.png\" style=\"vertical-align: -4px;\" title=\"Rendered by QuickLaTeX.com\" width=\"62\"\/> th\u1eb3ng h\u00e0ng).<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>III. D\u1ea1ng b\u00e0i t\u1eadp v\u1ec1 Ph\u00e9p to\u00e1n T\u1ecda \u0111\u1ed9 v\u00e0 T\u00edch V\u00f4 H\u01b0\u1edbng<\/strong><\/p>\n<p>\u0110\u00e2y l\u00e0 c\u00e1c b\u00e0i t\u1eadp s\u1eed d\u1ee5ng c\u00f4ng th\u1ee9c t\u1ecda \u0111\u1ed9 \u0111\u1ec3 gi\u1ea3i quy\u1ebft c\u00e1c v\u1ea5n \u0111\u1ec1 h\u00ecnh h\u1ecdc:<\/p>\n<ul class=\"wp-block-list\">\n<li><strong>Ph\u00e9p to\u00e1n Vect\u01a1 b\u1eb1ng T\u1ecda \u0111\u1ed9:<\/strong>\n<ul class=\"wp-block-list\">\n<li>T\u00ednh t\u1ecda \u0111\u1ed9 c\u1ee7a vect\u01a1 k\u1ebft qu\u1ea3 t\u1eeb c\u00e1c ph\u00e9p to\u00e1n tuy\u1ebfn t\u00ednh (v\u00ed d\u1ee5: <img loading=\"lazy\" decoding=\"async\" alt=\"\\vec{c} = 2\\vec{p} - 3\\vec{q} + \\vec{r}\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"17\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-4ad0f3361aa3dd4544fe32c07b122823_l3.png\" style=\"vertical-align: -4px;\" title=\"Rendered by QuickLaTeX.com\" width=\"121\"\/>).<\/li>\n<li>T\u00ecm tham s\u1ed1 <img loading=\"lazy\" decoding=\"async\" alt=\"m, n\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"12\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-5cff6dbbb18454fe1d42a419238876b7_l3.png\" style=\"vertical-align: -4px;\" title=\"Rendered by QuickLaTeX.com\" width=\"34\"\/> \u0111\u1ec3 hai vect\u01a1 <img loading=\"lazy\" decoding=\"async\" alt=\"\\vec{a}\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"13\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-2ef73196ac39b7d237c3a803bc5f7350_l3.png\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" width=\"11\"\/> v\u00e0 <img loading=\"lazy\" decoding=\"async\" alt=\"\\vec{b}\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"18\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-da71accbfb1191cd1ca030051704ba23_l3.png\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" width=\"10\"\/>\u00a0c\u00f9ng ph\u01b0\u01a1ng\u00a0(\u0111i\u1ec1u ki\u1ec7n <img loading=\"lazy\" decoding=\"async\" alt=\"\\frac{a_1}{b_1} = \\frac{a_2}{b_2} = \\frac{a_3}{b_3}\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"21\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-af963477090a48fab5ad0ca6f27eefda_l3.png\" style=\"vertical-align: -8px;\" title=\"Rendered by QuickLaTeX.com\" width=\"98\"\/>).<\/li>\n<\/ul>\n<\/li>\n<li><strong>T\u00edch V\u00f4 H\u01b0\u1edbng v\u00e0 G\u00f3c:<\/strong>\n<ul class=\"wp-block-list\">\n<li>T\u00ednh\u00a0T\u00edch v\u00f4 h\u01b0\u1edbng\u00a0<img loading=\"lazy\" decoding=\"async\" alt=\"\\vec{a} \\cdot \\vec{b}\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"18\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-bac25b8cb4745b957c2d0bbb0482dea9_l3.png\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" width=\"32\"\/>.<\/li>\n<li>X\u00e1c \u0111\u1ecbnh hai vect\u01a1\u00a0vu\u00f4ng g\u00f3c\u00a0(<img loading=\"lazy\" decoding=\"async\" alt=\"\\vec{a} \\perp \\vec{b} \\iff \\vec{a} \\cdot \\vec{b} = 0\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"19\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-4902ad255be44a94b8cfd4f3b09ff43a_l3.png\" style=\"vertical-align: -1px;\" title=\"Rendered by QuickLaTeX.com\" width=\"155\"\/>).<\/li>\n<li>T\u00ednh\u00a0\u0110\u1ed9 l\u1edbn\u00a0(modul) c\u1ee7a vect\u01a1 v\u00e0\u00a0\u0110\u1ed9 d\u00e0i\u00a0\u0111o\u1ea1n th\u1eb3ng <img loading=\"lazy\" decoding=\"async\" alt=\"AB\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"13\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-15c79f1659e7ab85acdfc229218e4393_l3-1.png\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" width=\"27\"\/>.<\/li>\n<li>T\u00ednh\u00a0G\u00f3c\u00a0gi\u1eefa hai vect\u01a1 <img loading=\"lazy\" decoding=\"async\" alt=\"\\cos(\\vec{a}, \\vec{b})\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"23\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-df730876abf12d77ebd9f10e4aaeca7b_l3.png\" style=\"vertical-align: -5px;\" title=\"Rendered by QuickLaTeX.com\" width=\"61\"\/>.<\/li>\n<\/ul>\n<\/li>\n<li><strong>\u1ee8ng d\u1ee5ng H\u00ecnh h\u1ecdc v\u00e0 C\u1ef1c tr\u1ecb:<\/strong>\n<ul class=\"wp-block-list\">\n<li>T\u00ecm t\u1ecda \u0111\u1ed9 \u0111\u1ec9nh c\u00f2n l\u1ea1i c\u1ee7a m\u1ed9t h\u00ecnh h\u00ecnh h\u1ecdc (v\u00ed d\u1ee5: h\u00ecnh b\u00ecnh h\u00e0nh, h\u00ecnh h\u1ed9p) khi bi\u1ebft t\u1ecda \u0111\u1ed9 c\u00e1c \u0111\u1ec9nh kh\u00e1c.<\/li>\n<li>T\u1ecda \u0111\u1ed9 h\u00f3a\u00a0m\u1ed9t h\u00ecnh kh\u00f4ng gian \u0111\u1ec3 gi\u1ea3i c\u00e1c b\u00e0i to\u00e1n v\u1ec1 kho\u1ea3ng c\u00e1ch, g\u00f3c, \u0111\u1ed9 d\u00e0i.<\/li>\n<li>B\u00e0i to\u00e1n\u00a0C\u1ef1c tr\u1ecb\u00a0(v\u00ed d\u1ee5: t\u00ecm \u0111i\u1ec3m <img loading=\"lazy\" decoding=\"async\" alt=\"M\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"12\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-279c67ecc79df39a3fb1845eddf29352_l3-1.png\" style=\"vertical-align: 0px;\" title=\"Rendered by QuickLaTeX.com\" width=\"19\"\/> \u0111\u1ec3 <img loading=\"lazy\" decoding=\"async\" alt=\"MA^2 + MB^2 + MC^2\" class=\"ql-img-inline-formula quicklatex-auto-format\" height=\"17\" src=\"https:\/\/77.buffdemo.com\/wp-content\/uploads\/2026\/04\/quicklatex-com-418a1f40977c6bd9d2b5797ec009016f_l3.png\" style=\"vertical-align: -2px;\" title=\"Rendered by QuickLaTeX.com\" width=\"165\"\/> \u0111\u1ea1t gi\u00e1 tr\u1ecb nh\u1ecf nh\u1ea5t).<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h2 class=\"wp-block-heading has-vivid-red-color has-text-color has-link-color wp-elements-4f6ebb75e1a921f643732ce1dbb13d01\" id=\"h-t\u1ea3i-tai-li\u1ec7u-toan-12-ch\u01b0\u01a1ng-2-vector-va-h\u1ec7-tr\u1ee5c-t\u1ecda-d\u1ed9-khong-gian\"><span class=\"ez-toc-section\" id=\"Tai_tai_lieu_Toan_12_Chuong_2_Vector_va_he_truc_toa_do_khong_gian\"><\/span>T\u1ea3i t\u00e0i li\u1ec7u To\u00e1n 12 Ch\u01b0\u01a1ng 2 Vector v\u00e0 h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 kh\u00f4ng gian<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>C\u00e1c b\u1ea1n c\u00f3 th\u1ec3 t\u1ea3i t\u00e0i li\u1ec7u \u00f4n t\u1eadp mi\u1ec5n ph\u00ed \u1edf \u0111\u00e2y nh\u00e9<\/p>\n<div class=\"wp-block-buttons is-content-justification-center is-layout-flex wp-container-core-buttons-is-layout-16018d1d wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link has-white-color has-luminous-vivid-orange-background-color has-text-color has-background has-link-color has-text-align-center wp-element-button\" href=\"https:\/\/drive.google.com\/drive\/folders\/1lgrid4IWSZa_GWlrN09rlOhcFtylzWMI?usp=sharing\" rel=\"noreferrer noopener nofollow\" target=\"_blank\"><strong>T\u1ea2I T\u00c0I LI\u1ec6U (\u0110\u1ec0 + \u0110\u00c1P \u00c1N)<\/strong><\/a><\/div>\n<\/div>\n<p>Tham kh\u1ea3o: <strong><a href=\"https:\/\/tailieuonthi.org\/toan-12-chuong-1-ung-dung-dao-ham-ham-so\/\" rel=\"noreferrer noopener\" target=\"_blank\">To\u00e1n 12 Ch\u01b0\u01a1ng 1: \u1ee8ng D\u1ee5ng \u0110\u1ea1o H\u00e0m \u0110\u1ec3 Kh\u1ea3o S\u00e1t V\u00e0 V\u1ebd \u0110\u1ed3 Th\u1ecb C\u1ee7a H\u00e0m S\u1ed1<\/a><\/strong><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Chia s\u1ebb t\u00e0i li\u1ec7u To\u00e1n 12 Ch\u01b0\u01a1ng 2: Vector v\u00e0 h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 kh\u00f4ng gian c\u00f3 l\u1eddi gi\u1ea3i chi ti\u1ebft, b\u00e1m s\u00e1t ch\u01b0\u01a1ng tr\u00ecnh m\u1edbi. N\u1ed9i dung t\u00e0i li\u1ec7u ch\u01b0\u01a1ng Vector v\u00e0 h\u1ec7 tr\u1ee5c t\u1ecda \u0111\u1ed9 kh\u00f4ng gian \u0110\u00e2y l\u00e0 m\u1ed9t b\u1ed9 t\u00e0i li\u1ec7u h\u1ecdc t\u1eadp to\u00e0n di\u1ec7n cho Ch\u01b0\u01a1ng II (Vect\u01a1 v\u00e0 H\u1ec7 [&#8230;]\n","protected":false},"author":1,"featured_media":2257,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[22,23,26],"tags":[],"class_list":["post-2283","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-tai-lieu-thpt","category-tai-lieu-toan","category-toan-12"],"_links":{"self":[{"href":"https:\/\/77.buffdemo.com\/index.php\/wp-json\/wp\/v2\/posts\/2283","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/77.buffdemo.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/77.buffdemo.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/77.buffdemo.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/77.buffdemo.com\/index.php\/wp-json\/wp\/v2\/comments?post=2283"}],"version-history":[{"count":0,"href":"https:\/\/77.buffdemo.com\/index.php\/wp-json\/wp\/v2\/posts\/2283\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/77.buffdemo.com\/index.php\/wp-json\/wp\/v2\/media\/2257"}],"wp:attachment":[{"href":"https:\/\/77.buffdemo.com\/index.php\/wp-json\/wp\/v2\/media?parent=2283"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/77.buffdemo.com\/index.php\/wp-json\/wp\/v2\/categories?post=2283"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/77.buffdemo.com\/index.php\/wp-json\/wp\/v2\/tags?post=2283"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}