{"id":406,"date":"2026-05-05T04:44:06","date_gmt":"2026-05-05T04:44:06","guid":{"rendered":"https:\/\/79.buffdemo.com\/index.php\/2026\/05\/05\/chuyen-de-quan-he-vuong-goc-trong-khong-gian-toan-11\/"},"modified":"2026-05-05T07:16:50","modified_gmt":"2026-05-05T07:16:50","slug":"chuyen-de-quan-he-vuong-goc-trong-khong-gian-toan-11","status":"publish","type":"post","link":"https:\/\/79.buffdemo.com\/index.php\/2026\/05\/05\/chuyen-de-quan-he-vuong-goc-trong-khong-gian-toan-11\/","title":{"rendered":"Chuy\u00ean \u0111\u1ec1 quan h\u1ec7 vu\u00f4ng g\u00f3c trong kh\u00f4ng gian To\u00e1n 11"},"content":{"rendered":"

Chuy\u00ean \u0111\u1ec1 Quan h\u1ec7 vu\u00f4ng g\u00f3c trong kh\u00f4ng gian To\u00e1n 11<\/strong> c\u1ee7a \u0111\u1ed9i ng\u0169 To\u00e1n T\u1eeb T\u00e2m l\u00e0 t\u00e0i li\u1ec7u \u00f4n t\u1eadp chuy\u00ean s\u00e2u, gi\u00fap h\u1ecdc sinh l\u1edbp 11 n\u1eafm v\u1eefng ki\u1ebfn th\u1ee9c v\u1ec1 quan h\u1ec7 vu\u00f4ng g\u00f3c trong h\u00ecnh h\u1ecdc kh\u00f4ng gian. V\u1edbi h\u1ec7 th\u1ed1ng b\u00e0i gi\u1ea3ng chi ti\u1ebft, b\u00e0i t\u1eadp \u0111a d\u1ea1ng t\u1eeb tr\u1eafc nghi\u1ec7m \u0111\u1ebfn t\u1ef1 lu\u1eadn, \u0111\u00e2y ch\u00ednh l\u00e0 t\u00e0i li\u1ec7u gi\u00fap b\u1ea1n l\u00e0m ch\u1ee7 c\u00e1c d\u1ea1ng to\u00e1n quan tr\u1ecdng. H\u00e3y c\u00f9ng T\u00e0i Li\u1ec7u \u00d4n Thi<\/strong><\/a> kh\u00e1m ph\u00e1 ngay b\u1ed9 t\u00e0i li\u1ec7u n\u00e0y nh\u00e9.<\/p>\n

N\u1ed9i dung Chuy\u00ean \u0111\u1ec1 Quan h\u1ec7 vu\u00f4ng g\u00f3c trong kh\u00f4ng gian To\u00e1n 11<\/strong><\/h2>\n

Chuy\u00ean \u0111\u1ec1 Quan h\u1ec7 vu\u00f4ng g\u00f3c trong kh\u00f4ng gian To\u00e1n 11<\/strong> l\u00e0 t\u00e0i li\u1ec7u gi\u00fap h\u1ecdc sinh n\u1eafm v\u1eefng c\u00e1c ki\u1ebfn th\u1ee9c v\u00e0 k\u1ef9 n\u0103ng li\u00ean quan \u0111\u1ebfn quan h\u1ec7 vu\u00f4ng g\u00f3c trong h\u00ecnh h\u1ecdc kh\u00f4ng gian. C\u1ea5u tr\u00fac t\u00e0i li\u1ec7u g\u1ed3m 5 b\u00e0i l\u1edbn, trong m\u1ed7i b\u00e0i g\u1ed3m c\u00f3 3 ph\u1ea7n: A. L\u00fd thuy\u1ebft, B. C\u00e1c d\u1ea1ng b\u00e0i t\u1eadp, C. Luy\u1ec7n t\u1eadp.<\/p>\n

    \n
  • B\u00e0i 1: Hai \u0111\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c<\/li>\n
  • B\u00e0i 2: \u0110\u01b0\u1eddng th\u1eb3ng vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng<\/li>\n
  • B\u00e0i 3: Hai m\u1eb7t ph\u1eb3ng vu\u00f4ng g\u00f3c<\/li>\n
  • B\u00e0i 4: Kho\u1ea3ng c\u00e1ch & th\u1ec3 t\u00edch<\/li>\n
  • B\u00e0i 5: G\u00f3c gi\u1eefa \u0111\u01b0\u1eddng th\u1eb3ng \u2013 m\u1eb7t ph\u1eb3ng & g\u00f3c nh\u1ecb di\u1ec7n<\/li>\n<\/ul>\n

    Tr\u00edch m\u1ed9t s\u1ed1 v\u00ed d\u1ee5 v\u00e0 c\u00e2u h\u1ecfi trong t\u00e0i li\u1ec7u to\u00e1n l\u1edbp 11 Chuy\u00ean \u0111\u1ec1 Quan h\u1ec7 vu\u00f4ng g\u00f3c trong kh\u00f4ng gian To\u00e1n 11<\/strong><\/p>\n

    V\u00ed d\u1ee5 1.1.<\/strong> Cho h\u00ecnh l\u1eadp ph\u01b0\u01a1ng ABCD.A\u2081B\u2081C\u2081D\u2081 (h\u00ecnh v\u1ebd b\u00ean). X\u00e1c \u0111\u1ecbnh g\u00f3c gi\u1eefa c\u00e1c c\u1eb7p \u0111\u01b0\u1eddng th\u1eb3ng: A\u2081C\u2081 v\u00e0 AB , B\u2081C v\u00e0 BD .<\/p>\n

    V\u00ed d\u1ee5 1.2.<\/strong> Cho h\u00ecnh ch\u00f3p S.ABCD c\u00f3 \u0111\u00e1y l\u00e0 h\u00ecnh thoi, SA = AB v\u00e0 SA perp BC . T\u00ednh g\u00f3c gi\u1eefa hai \u0111\u01b0\u1eddng th\u1eb3ng SD v\u00e0 BC .<\/p>\n

    V\u00ed d\u1ee5 1.3.<\/strong> Cho h\u00ecnh l\u1eadp ph\u01b0\u01a1ng ABCD.A\u2019B\u2019C\u2019D\u2019 . T\u00ednh g\u00f3c gi\u1eefa hai \u0111\u01b0\u1eddng th\u1eb3ng: AB v\u00e0 B\u2019C\u2019 , AC v\u00e0 B\u2019C\u2019 , A\u2019C\u2019 v\u00e0 B\u2019C\u2019 .<\/p>\n

    V\u00ed d\u1ee5 1.4.<\/strong> Cho h\u00ecnh ch\u00f3p S.ABCD c\u00f3 \u0111\u00e1y l\u00e0 h\u00ecnh vu\u00f4ng ABCD c\u1ea1nh b\u1eb1ng a v\u00e0 c\u00e1c c\u1ea1nh b\u00ean \u0111\u1ec1u b\u1eb1ng a . G\u1ecdi M v\u00e0 N l\u1ea7n l\u01b0\u1ee3t l\u00e0 trung \u0111i\u1ec3m c\u1ee7a AD v\u00e0 SD . S\u1ed1 \u0111o c\u1ee7a g\u00f3c (MN, SC) b\u1eb1ng bao nhi\u00eau?<\/p>\n

    C\u00e2u 1.<\/strong> Trong kh\u00f4ng gian, g\u00f3c gi\u1eefa hai \u0111\u01b0\u1eddng th\u1eb3ng a v\u00e0 b l\u00e0 g\u00f3c gi\u1eefa hai \u0111\u01b0\u1eddng th\u1eb3ng a\u2019 v\u00e0 b\u2019 th\u1ecfa m\u00e3n<\/p>\n

    A. C\u00f9ng \u0111i qua m\u1ed9t \u0111i\u1ec3m v\u00e0 l\u1ea7n l\u01b0\u1ee3t song song ho\u1eb7c tr\u00f9ng v\u1edbi a v\u00e0 b .
    \nB. L\u1ea7n l\u01b0\u1ee3t song song ho\u1eb7c tr\u00f9ng v\u1edbi a v\u00e0 b .
    \nC. C\u00f9ng \u0111i qua m\u1ed9t \u0111i\u1ec3m v\u00e0 l\u1ea7n l\u01b0\u1ee3t song song v\u1edbi a v\u00e0 b .
    \nD. L\u1ea7n l\u01b0\u1ee3t song song v\u1edbi a v\u00e0 b .<\/p>\n

    C\u00e2u 2.<\/strong> Trong kh\u00f4ng gian, hai \u0111\u01b0\u1eddng th\u1eb3ng a v\u00e0 b \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 vu\u00f4ng g\u00f3c v\u1edbi nhau n\u1ebfu g\u00f3c gi\u1eefa ch\u00fang b\u1eb1ng A. 90^\u00b0 . B. 45^\u00b0 . C. 0^\u00b0 . D. 180^\u00b0 .<\/p>\n

    T\u1ea3i Chuy\u00ean \u0111\u1ec1 quan h\u1ec7 vu\u00f4ng g\u00f3c trong kh\u00f4ng gian To\u00e1n 11<\/strong><\/h2>\n

    Chuy\u00ean \u0111\u1ec1 Quan h\u1ec7 vu\u00f4ng g\u00f3c trong kh\u00f4ng gian To\u00e1n 11<\/strong> s\u1ebd gi\u00fap b\u1ea1n t\u1ef1 tin gi\u1ea3i quy\u1ebft m\u1ecdi d\u1ea1ng b\u00e0i t\u1eadp t\u1eeb c\u01a1 b\u1ea3n \u0111\u1ebfn n\u00e2ng cao. V\u1edbi h\u1ec7 th\u1ed1ng l\u00fd thuy\u1ebft r\u00f5 r\u00e0ng c\u00f9ng ph\u01b0\u01a1ng ph\u00e1p gi\u1ea3i chi ti\u1ebft, \u0111\u00e2y ch\u00ednh l\u00e0 ch\u00eca kh\u00f3a gi\u00fap b\u1ea1n \u0111\u1ea1t \u0111i\u1ec3m s\u1ed1 cao h\u01a1n. \u0110\u1eebng b\u1ecf l\u1ee1, t\u1ea3i ngay t\u00e0i li\u1ec7u theo \u0111\u01b0\u1eddng d\u1eabn b\u00ean d\u01b0\u1edbi nh\u00e9!<\/p>\n