{"id":567213,"date":"2026-07-22T15:20:04","date_gmt":"2026-07-22T08:20:04","guid":{"rendered":"https:\/\/hoanghamobile.com\/tin-tuc\/?p=567213"},"modified":"2026-07-22T15:20:04","modified_gmt":"2026-07-22T08:20:04","slug":"phuong-trinh-mat-phang","status":"publish","type":"post","link":"https:\/\/hoanghamobile.com\/tin-tuc\/phuong-trinh-mat-phang\/","title":{"rendered":"Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng: c\u00f4ng th\u1ee9c, c\u00e1ch vi\u1ebft v\u00e0 v\u00ed d\u1ee5"},"content":{"rendered":"

Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng m\u00f4 t\u1ea3 m\u1ed9t m\u1eb7t ph\u1eb3ng trong kh\u00f4ng gian Oxyz b\u1eb1ng m\u1ed9t h\u1ec7 th\u1ee9c \u0111\u1ea1i s\u1ed1 duy nh\u1ea5t, thay v\u00ec ph\u1ea3i h\u00ecnh dung qua h\u00ecnh v\u1ebd. Ch\u1ec9 v\u1edbi b\u1ed1n h\u1ec7 s\u1ed1 a, b, c, d, b\u1ea1n x\u00e1c \u0111\u1ecbnh \u0111\u01b0\u1ee3c m\u1eb7t ph\u1eb3ng nghi\u00eang theo h\u01b0\u1edbng n\u00e0o, c\u00f3 song song hay vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng kh\u00e1c hay kh\u00f4ng, v\u00e0 t\u00ednh \u0111\u01b0\u1ee3c kho\u1ea3ng c\u00e1ch t\u1eeb m\u1ed9t \u0111i\u1ec3m b\u1ea5t k\u1ef3 \u0111\u1ebfn n\u00f3. \u0110\u00e2y l\u00e0 ki\u1ebfn th\u1ee9c n\u1ec1n c\u1ee7a ch\u01b0\u01a1ng h\u00ecnh h\u1ecdc kh\u00f4ng gian Oxyz l\u1edbp 12, xu\u1ea5t hi\u1ec7n trong h\u1ea7u h\u1ebft \u0111\u1ec1 thi t\u1ed1t nghi\u1ec7p THPT. N\u1ed9i dung sau t\u1ed5ng h\u1ee3p c\u00f4ng th\u1ee9c, c\u00e1c c\u00e1ch vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng ph\u1ed5 bi\u1ebfn nh\u1ea5t, c\u00f9ng v\u00ed d\u1ee5 c\u00f3 l\u1eddi gi\u1ea3i chi ti\u1ebft.<\/p>\n

Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng l\u00e0 g\u00ec?<\/h2>\n

Trong kh\u00f4ng gian Oxyz, m\u1ed7i m\u1eb7t ph\u1eb3ng \u0111\u1ec1u bi\u1ec3u di\u1ec5n \u0111\u01b0\u1ee3c b\u1eb1ng m\u1ed9t ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc nh\u1ea5t ba \u1ea9n x, y, z.<\/p>\n

Ph\u01b0\u01a1ng tr\u00ecnh t\u1ed5ng qu\u00e1t c\u1ee7a m\u1eb7t ph\u1eb3ng (P)<\/strong> c\u00f3 d\u1ea1ng:<\/p>\n

ax + by + cz + d = 0, trong \u0111\u00f3 a, b, c kh\u00f4ng \u0111\u1ed3ng th\u1eddi b\u1eb1ng 0 (t\u1ee9c a\u00b2 + b\u00b2 + c\u00b2 > 0).<\/p>\n

B\u1ed9 ba s\u1ed1 (a; b; c) ch\u00ednh l\u00e0 t\u1ecda \u0111\u1ed9 c\u1ee7a vect\u01a1 ph\u00e1p tuy\u1ebfn \u2014 vect\u01a1 vu\u00f4ng g\u00f3c v\u1edbi m\u1ecdi vect\u01a1 n\u1eb1m trong m\u1eb7t ph\u1eb3ng \u0111\u00f3. H\u1ec7 s\u1ed1 d quy\u1ebft \u0111\u1ecbnh m\u1eb7t ph\u1eb3ng c\u00e1ch g\u1ed1c t\u1ecda \u0111\u1ed9 bao xa; khi d = 0, m\u1eb7t ph\u1eb3ng \u0111i qua g\u1ed1c t\u1ecda \u0111\u1ed9 O(0; 0; 0).<\/p>\n

\"phuong-trinh-mat-phang-1\"<\/p>\n

Vect\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a m\u1eb7t ph\u1eb3ng<\/h2>\n

Vect\u01a1 ph\u00e1p tuy\u1ebfn l\u00e0 vect\u01a1 kh\u00e1c 0 c\u00f3 gi\u00e1 vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng.<\/p>\n

N\u1ebfu m\u1eb7t ph\u1eb3ng (P) c\u00f3 ph\u01b0\u01a1ng tr\u00ecnh ax + by + cz + d = 0 th\u00ec n = (a; b; c) l\u00e0 m\u1ed9t vect\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a (P). M\u1ed9t m\u1eb7t ph\u1eb3ng c\u00f3 v\u00f4 s\u1ed1 vect\u01a1 ph\u00e1p tuy\u1ebfn, t\u1ea5t c\u1ea3 c\u00f9ng ph\u01b0\u01a1ng v\u1edbi nhau \u2014 n\u1ebfu n l\u00e0 m\u1ed9t vect\u01a1 ph\u00e1p tuy\u1ebfn th\u00ec k\u00b7n (k \u2260 0) c\u0169ng l\u00e0 vect\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a m\u1eb7t ph\u1eb3ng \u0111\u00f3.<\/p>\n

Nhi\u1ec1u b\u00e0i kh\u00f4ng cho s\u1eb5n vect\u01a1 ph\u00e1p tuy\u1ebfn m\u00e0 ph\u1ea3i suy ra t\u1eeb hai vect\u01a1 n\u1eb1m trong m\u1eb7t ph\u1eb3ng, b\u1eb1ng t\u00edch c\u00f3 h\u01b0\u1edbng \u2014 c\u00e1ch l\u00e0m c\u1ee5 th\u1ec3 \u1edf ph\u1ea7n v\u00ed d\u1ee5 b\u00ean d\u01b0\u1edbi.<\/p>\n

\"phuong-trinh-mat-phang-2\"<\/p>\n

C\u00e1c d\u1ea1ng ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng th\u01b0\u1eddng g\u1eb7p<\/h2>\n

T\u00f9y d\u1eef ki\u1ec7n \u0111\u1ec1 b\u00e0i, b\u1ea1n vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng theo ba d\u1ea1ng ph\u1ed5 bi\u1ebfn sau.<\/p>\n\n\n\n\n\n\n\n
D\u1ea1ng<\/th>\nC\u00f4ng th\u1ee9c<\/th>\n\u00c1p d\u1ee5ng khi<\/th>\n<\/tr>\n<\/thead>\n
T\u1ed5ng qu\u00e1t<\/td>\nax + by + cz + d = 0<\/td>\n\u0110\u00e3 bi\u1ebft vect\u01a1 ph\u00e1p tuy\u1ebfn (a; b; c)<\/td>\n<\/tr>\n
Qua 1 \u0111i\u1ec3m + ph\u00e1p tuy\u1ebfn<\/td>\na(x – x0) + b(y – y0) + c(z – z0) = 0<\/td>\nBi\u1ebft \u0111i\u1ec3m M0(x0; y0; z0) v\u00e0 vect\u01a1 ph\u00e1p tuy\u1ebfn n = (a; b; c)<\/td>\n<\/tr>\n
\u0110o\u1ea1n ch\u1eafn<\/td>\nx\/p + y\/q + z\/r = 1<\/td>\nBi\u1ebft 3 giao \u0111i\u1ec3m v\u1edbi 3 tr\u1ee5c t\u1ecda \u0111\u1ed9: (p;0;0), (0;q;0), (0;0;r), p, q, r \u2260 0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

D\u1ea1ng th\u1ee9 hai d\u00f9ng nhi\u1ec1u nh\u1ea5t v\u00ec h\u1ea7u h\u1ebft \u0111\u1ec1 cho s\u1eb5n m\u1ed9t \u0111i\u1ec3m thu\u1ed9c m\u1eb7t ph\u1eb3ng c\u00f9ng vect\u01a1 ph\u00e1p tuy\u1ebfn. L\u01b0u \u00fd p, q, r trong d\u1ea1ng \u0111o\u1ea1n ch\u1eafn l\u00e0 \u0111o\u1ea1n ch\u1eafn tr\u00ean t\u1eebng tr\u1ee5c, kh\u00e1c v\u1edbi h\u1ec7 s\u1ed1 a, b, c c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh t\u1ed5ng qu\u00e1t.<\/p>\n

\"phuong-trinh-mat-phang-3\"<\/p>\n

C\u00e1ch vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng qua 3 \u0111i\u1ec3m kh\u00f4ng th\u1eb3ng h\u00e0ng<\/h2>\n

Khi \u0111\u1ec1 cho 3 \u0111i\u1ec3m A, B, C kh\u00f4ng th\u1eb3ng h\u00e0ng, b\u1ea1n t\u00ecm vect\u01a1 ph\u00e1p tuy\u1ebfn b\u1eb1ng t\u00edch c\u00f3 h\u01b0\u1edbng c\u1ee7a hai vect\u01a1 t\u1ea1o t\u1eeb 3 \u0111i\u1ec3m, theo 3 b\u01b0\u1edbc sau.<\/p>\n

B\u01b0\u1edbc 1:<\/strong> L\u1eadp hai vect\u01a1 AB v\u00e0 AC t\u1eeb t\u1ecda \u0111\u1ed9 3 \u0111i\u1ec3m.<\/p>\n

B\u01b0\u1edbc 2:<\/strong> T\u00ednh t\u00edch c\u00f3 h\u01b0\u1edbng n = [AB, AC] \u2014 \u0111\u00e2y l\u00e0 vect\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a m\u1eb7t ph\u1eb3ng (ABC).<\/p>\n

B\u01b0\u1edbc 3:<\/strong> Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng qua A (ho\u1eb7c B, C) v\u1edbi ph\u00e1p tuy\u1ebfn n v\u1eeba t\u00ecm, theo c\u00f4ng th\u1ee9c a(x – x0) + b(y – y0) + c(z – z0) = 0.<\/p>\n

C\u00e1ch n\u00e0y \u00e1p d\u1ee5ng cho m\u1ecdi tr\u01b0\u1eddng h\u1ee3p 3 \u0111i\u1ec3m kh\u00f4ng th\u1eb3ng h\u00e0ng, k\u1ec3 c\u1ea3 khi \u0111\u1ec1 kh\u00f4ng cho s\u1eb5n vect\u01a1 ph\u00e1p tuy\u1ebfn.<\/p>\n

\"phuong-trinh-mat-phang-4\"<\/p>\n

C\u00f4ng th\u1ee9c t\u00ednh kho\u1ea3ng c\u00e1ch t\u1eeb m\u1ed9t \u0111i\u1ec3m \u0111\u1ebfn m\u1eb7t ph\u1eb3ng<\/h2>\n

Bi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng, b\u1ea1n t\u00ednh \u0111\u01b0\u1ee3c ngay kho\u1ea3ng c\u00e1ch t\u1eeb b\u1ea5t k\u1ef3 \u0111i\u1ec3m n\u00e0o \u0111\u1ebfn m\u1eb7t ph\u1eb3ng m\u00e0 kh\u00f4ng c\u1ea7n d\u1ef1ng h\u00ecnh.<\/p>\n

Cho m\u1eb7t ph\u1eb3ng (P): ax + by + cz + d = 0 v\u00e0 \u0111i\u1ec3m M0(x0; y0; z0). Kho\u1ea3ng c\u00e1ch t\u1eeb M0 \u0111\u1ebfn (P) l\u00e0:<\/p>\n

d(M0, (P)) = |a\u00b7x0 + b\u00b7y0 + c\u00b7z0 + d| \/ \u221a(a\u00b2 + b\u00b2 + c\u00b2)<\/p>\n

C\u00f4ng th\u1ee9c d\u00f9ng d\u1ea5u gi\u00e1 tr\u1ecb tuy\u1ec7t \u0111\u1ed1i \u1edf t\u1eed s\u1ed1 v\u00ec kho\u1ea3ng c\u00e1ch lu\u00f4n kh\u00f4ng \u00e2m, b\u1ea5t k\u1ec3 \u0111i\u1ec3m n\u1eb1m v\u1ec1 ph\u00eda n\u00e0o c\u1ee7a m\u1eb7t ph\u1eb3ng.<\/p>\n

V\u00ed d\u1ee5 v\u1ec1 ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng c\u00f3 l\u1eddi gi\u1ea3i<\/h2>\n

V\u00ed d\u1ee5 1.<\/strong> Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (P) qua \u0111i\u1ec3m M(1; 2; -3), nh\u1eadn n = (2; -1; 4) l\u00e0m vect\u01a1 ph\u00e1p tuy\u1ebfn.<\/p>\n

L\u1eddi gi\u1ea3i:<\/em> \u00c1p d\u1ee5ng a(x – x0) + b(y – y0) + c(z – z0) = 0 v\u1edbi (a; b; c) = (2; -1; 4), M0 = M(1; 2; -3):<\/p>\n

2(x – 1) – 1(y – 2) + 4(z + 3) = 0 \u21d4 2x – 2 – y + 2 + 4z + 12 = 0 \u21d4 2x – y + 4z + 12 = 0.<\/p>\n

V\u1eady (P): 2x – y + 4z + 12 = 0.<\/p>\n

V\u00ed d\u1ee5 2.<\/strong> Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng (ABC) bi\u1ebft A(1; 0; 1), B(2; 1; 0), C(0; 2; 1).<\/p>\n

L\u1eddi gi\u1ea3i:<\/em><\/p>\n

B\u01b0\u1edbc 1:<\/strong> AB = (1; 1; -1), AC = (-1; 2; 0).<\/p>\n

B\u01b0\u1edbc 2:<\/strong> n = [AB, AC] = (1\u00b70 – (-1)\u00b72; (-1)\u00b7(-1) – 1\u00b70; 1\u00b72 – 1\u00b7(-1)) = (2; 1; 3).<\/p>\n

B\u01b0\u1edbc 3:<\/strong> Qua A(1; 0; 1) v\u1edbi ph\u00e1p tuy\u1ebfn n = (2; 1; 3): 2(x – 1) + 1(y – 0) + 3(z – 1) = 0 \u21d4 2x + y + 3z – 5 = 0.<\/p>\n

V\u1eady (ABC): 2x + y + 3z – 5 = 0. B\u1ea1n c\u00f3 th\u1ec3 thay B, C v\u00e0o \u0111\u1ec3 ki\u1ec3m tra \u2014 c\u1ea3 hai \u0111\u1ec1u th\u1ecfa m\u00e3n.<\/p>\n

V\u00ed d\u1ee5 3.<\/strong> T\u00ednh kho\u1ea3ng c\u00e1ch t\u1eeb M(1; 2; 3) \u0111\u1ebfn m\u1eb7t ph\u1eb3ng (P): 2x + y + 3z – 5 = 0 (m\u1eb7t ph\u1eb3ng (ABC) \u1edf v\u00ed d\u1ee5 2).<\/p>\n

L\u1eddi gi\u1ea3i:<\/em> d(M, (P)) = |2\u00b71 + 1\u00b72 + 3\u00b73 – 5| \/ \u221a(2\u00b2 + 1\u00b2 + 3\u00b2) = 8\/\u221a14 \u2248 2,14.<\/p>\n

\"phuong-trinh-mat-phang-5\"<\/p>\n

B\u00e0i t\u1eadp t\u1ef1 luy\u1ec7n<\/h2>\n

B\u1ea1n th\u1eed \u00e1p d\u1ee5ng c\u00f4ng th\u1ee9c v\u1eeba h\u1ecdc \u0111\u1ec3 gi\u1ea3i hai b\u00e0i sau, r\u1ed3i so \u0111\u00e1p s\u1ed1.<\/p>\n

B\u00e0i 1.<\/strong> Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng qua N(1; 0; 0), song song v\u1edbi (Q): x – 2y + 3z – 4 = 0.<\/p>\n

G\u1ee3i \u00fd:<\/em> hai m\u1eb7t ph\u1eb3ng song song c\u00f3 c\u00f9ng vect\u01a1 ph\u00e1p tuy\u1ebfn. \u0110\u00e1p s\u1ed1:<\/em> x – 2y + 3z – 1 = 0.<\/p>\n

B\u00e0i 2.<\/strong> T\u00ednh kho\u1ea3ng c\u00e1ch t\u1eeb g\u1ed1c t\u1ecda \u0111\u1ed9 O \u0111\u1ebfn m\u1eb7t ph\u1eb3ng (R): 3x – 4y + 12z – 26 = 0.<\/p>\n

G\u1ee3i \u00fd:<\/em> \u00e1p d\u1ee5ng tr\u1ef1c ti\u1ebfp c\u00f4ng th\u1ee9c kho\u1ea3ng c\u00e1ch v\u1edbi M0 = O(0; 0; 0). \u0110\u00e1p s\u1ed1:<\/em> d = 2.<\/p>\n

L\u1ed7i th\u01b0\u1eddng g\u1eb7p khi vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng<\/h2>\n

V\u00e0i l\u1ed7i l\u1eb7p l\u1ea1i nhi\u1ec1u nh\u1ea5t khi l\u00e0m d\u1ea1ng b\u00e0i n\u00e0y:<\/p>\n

    \n
  • Nh\u1ea7m t\u1ecda \u0111\u1ed9 vect\u01a1 ph\u00e1p tuy\u1ebfn v\u1edbi t\u1ecda \u0111\u1ed9 \u0111i\u1ec3m khi th\u1ebf v\u00e0o c\u00f4ng th\u1ee9c.<\/li>\n
  • Qu\u00ean \u0111i\u1ec1u ki\u1ec7n a\u00b2 + b\u00b2 + c\u00b2 > 0 \u2014 n = (0; 0; 0) ngh\u0129a l\u00e0 3 \u0111i\u1ec3m \u0111\u00e3 cho th\u1eb3ng h\u00e0ng, kh\u00f4ng l\u1eadp \u0111\u01b0\u1ee3c m\u1eb7t ph\u1eb3ng.<\/li>\n
  • Nh\u1ea7m d\u1ea5u khi khai tri\u1ec3n a(x – x0), nh\u1ea5t l\u00e0 khi x0 \u00e2m.<\/li>\n
  • B\u1ecf d\u1ea5u gi\u00e1 tr\u1ecb tuy\u1ec7t \u0111\u1ed1i \u1edf c\u00f4ng th\u1ee9c kho\u1ea3ng c\u00e1ch \u2014 kho\u1ea3ng c\u00e1ch kh\u00f4ng th\u1ec3 \u00e2m.<\/li>\n<\/ul>\n

    T\u01b0\u01a1ng t\u1ef1 c\u00e1ch vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng<\/a>, ph\u1ea7n d\u1ec5 sai nh\u1ea5t l\u00e0 x\u00e1c \u0111\u1ecbnh \u0111\u00fang vect\u01a1 \u0111\u1eb7c tr\u01b0ng tr\u01b0\u1edbc khi th\u1ebf c\u00f4ng th\u1ee9c.<\/p>\n

    C\u00e2u h\u1ecfi th\u01b0\u1eddng g\u1eb7p v\u1ec1 ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng<\/h2>\n

    Ph\u01b0\u01a1ng tr\u00ecnh t\u1ed5ng qu\u00e1t c\u1ee7a m\u1eb7t ph\u1eb3ng l\u00e0 g\u00ec?<\/h3>\n

    L\u00e0 ph\u01b0\u01a1ng tr\u00ecnh ax + by + cz + d = 0, v\u1edbi a, b, c kh\u00f4ng \u0111\u1ed3ng th\u1eddi b\u1eb1ng 0. B\u1ed9 (a; b; c) l\u00e0 t\u1ecda \u0111\u1ed9 vect\u01a1 ph\u00e1p tuy\u1ebfn.<\/p>\n

    Vect\u01a1 ph\u00e1p tuy\u1ebfn c\u1ee7a m\u1eb7t ph\u1eb3ng l\u00e0 g\u00ec?<\/h3>\n

    L\u00e0 vect\u01a1 kh\u00e1c 0 c\u00f3 gi\u00e1 vu\u00f4ng g\u00f3c v\u1edbi m\u1eb7t ph\u1eb3ng. N\u1ebfu (P): ax + by + cz + d = 0 th\u00ec n = (a; b; c) l\u00e0 m\u1ed9t ph\u00e1p tuy\u1ebfn c\u1ee7a (P).<\/p>\n

    C\u00e1ch vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng qua m\u1ed9t \u0111i\u1ec3m v\u00e0 vect\u01a1 ph\u00e1p tuy\u1ebfn?<\/h3>\n

    D\u00f9ng c\u00f4ng th\u1ee9c a(x – x0) + b(y – y0) + c(z – z0) = 0, v\u1edbi M0(x0; y0; z0) l\u00e0 \u0111i\u1ec3m m\u1eb7t ph\u1eb3ng \u0111i qua, n = (a; b; c) l\u00e0 ph\u00e1p tuy\u1ebfn.<\/p>\n

    Khi n\u00e0o m\u1eb7t ph\u1eb3ng \u0111i qua g\u1ed1c t\u1ecda \u0111\u1ed9?<\/h3>\n

    Khi d = 0 trong ph\u01b0\u01a1ng tr\u00ecnh t\u1ed5ng qu\u00e1t. L\u00fac \u0111\u00f3 O(0; 0; 0) lu\u00f4n th\u1ecfa m\u00e3n ph\u01b0\u01a1ng tr\u00ecnh.<\/p>\n

    M\u1ed9t m\u1eb7t ph\u1eb3ng c\u00f3 bao nhi\u00eau vect\u01a1 ph\u00e1p tuy\u1ebfn?<\/h3>\n

    C\u00f3 v\u00f4 s\u1ed1 \u2014 m\u1ecdi vect\u01a1 c\u00f9ng ph\u01b0\u01a1ng v\u1edbi m\u1ed9t ph\u00e1p tuy\u1ebfn \u0111\u00e3 bi\u1ebft, t\u1ee9c nh\u00e2n v\u1edbi k \u2260 0 b\u1ea5t k\u1ef3, \u0111\u1ec1u l\u00e0 ph\u00e1p tuy\u1ebfn c\u1ee7a c\u00f9ng m\u1eb7t ph\u1eb3ng.<\/p>\n

    L\u00e0m sao vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng khi bi\u1ebft 3 \u0111i\u1ec3m kh\u00f4ng th\u1eb3ng h\u00e0ng?<\/h3>\n

    L\u1eadp hai vect\u01a1 t\u1eeb 3 \u0111i\u1ec3m, l\u1ea5y t\u00edch c\u00f3 h\u01b0\u1edbng \u0111\u1ec3 \u0111\u01b0\u1ee3c vect\u01a1 ph\u00e1p tuy\u1ebfn, r\u1ed3i vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh qua m\u1ed9t trong 3 \u0111i\u1ec3m v\u1edbi ph\u00e1p tuy\u1ebfn v\u1eeba t\u00ecm.<\/p>\n

    D\u1ea1ng ph\u01b0\u01a1ng tr\u00ecnh \u0111o\u1ea1n ch\u1eafn c\u1ee7a m\u1eb7t ph\u1eb3ng d\u00f9ng khi n\u00e0o?<\/h3>\n

    Khi c\u1ea7n bi\u1ec3u di\u1ec5n theo 3 giao \u0111i\u1ec3m c\u1ee7a m\u1eb7t ph\u1eb3ng v\u1edbi 3 tr\u1ee5c t\u1ecda \u0111\u1ed9: x\/p + y\/q + z\/r = 1, v\u1edbi p, q, r l\u00e0 \u0111o\u1ea1n ch\u1eafn kh\u00e1c 0 tr\u00ean t\u1eebng tr\u1ee5c.<\/p>\n

    C\u00f4ng th\u1ee9c t\u00ednh kho\u1ea3ng c\u00e1ch t\u1eeb m\u1ed9t \u0111i\u1ec3m \u0111\u1ebfn m\u1eb7t ph\u1eb3ng l\u00e0 g\u00ec?<\/h3>\n

    d(M0, (P)) = |a\u00b7x0 + b\u00b7y0 + c\u00b7z0 + d| \/ \u221a(a\u00b2 + b\u00b2 + c\u00b2), v\u1edbi (P): ax + by + cz + d = 0.<\/p>\n

    K\u1ebft lu\u1eadn<\/h2>\n

    Ph\u01b0\u01a1ng tr\u00ecnh m\u1eb7t ph\u1eb3ng g\u00f3i g\u1ecdn trong m\u1ed9t h\u1ec7 th\u1ee9c b\u1eadc nh\u1ea5t ax + by + cz + d = 0, nh\u01b0ng \u0111\u1ee7 \u0111\u1ec3 gi\u1ea3i h\u1ea7u h\u1ebft b\u00e0i to\u00e1n kh\u00f4ng gian Oxyz: vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh qua \u0111i\u1ec3m, qua 3 \u0111i\u1ec3m, hay t\u00ednh kho\u1ea3ng c\u00e1ch. Ch\u00eca kh\u00f3a l\u00e0 lu\u00f4n x\u00e1c \u0111\u1ecbnh \u0111\u00fang vect\u01a1 ph\u00e1p tuy\u1ebfn tr\u01b0\u1edbc khi th\u1ebf v\u00e0o c\u00f4ng th\u1ee9c. B\u1ea1n n\u00ean luy\u1ec7n th\u00eam v\u00e0i b\u00e0i t\u1ef1 luy\u1ec7n \u0111\u1ec3 quen tay, \u0111\u1ed3ng th\u1eddi c\u00f3 th\u1ec3 \u00f4n th\u00eam ph\u01b0\u01a1ng tr\u00ecnh ch\u00ednh t\u1eafc c\u1ee7a elip<\/a> \u2014 m\u1ed9t d\u1ea1ng ph\u01b0\u01a1ng tr\u00ecnh kh\u00e1c c\u0169ng hay g\u1eb7p trong \u0111\u1ec1 thi.<\/p>\n