Trong không gian Oxyz, cho đường thẳng \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaacQ % dadaWcaaqaaiaadIhacqGHsislcaaIZaaabaGaaGOmaaaacqGH9aqp % daWcaaqaaiaadMhacqGHsislcaaI0aaabaGaaGymaaaacqGH9aqpda % WcaaqaaiaadQhacqGHsislcaaIYaaabaGaaGymaaaaaaa!4401! d:\frac{{x - 3}}{2} = \frac{{y - 4}}{1} = \frac{{z - 2}}{1}\) và 2 điểm A( 6;3;-2); B(1;0;-1). Gọi \(\Delta\) là đường thẳng đi qua B, vuông góc với d và thỏa mãn khoảng cách từ A đến \(\Delta\) là nhỏ nhất. Một vectơ chỉ phương của có tọa độ:

A. (1;1;-3)

B. (1;-1;-1)

C. (1;2;-4)

D. (2;-1;-3)

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Chủ đề: Đề thi THPT QG

Môn: Toán

Lời giải:

Báo sai

Gọi (P) là mặt phẳng đi qua B và vuông góc với \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabgk % DiEpaabmaabaGaamiuaaGaayjkaiaawMcaaiaacQdacaaIYaGaamiE % aiabgUcaRiaadMhacqGHRaWkcaWG6bGaeyOeI0IaaGymaiabg2da9i % aaicdacaGGUaGaaGPaVlaaykW7caaMc8oaaa!4A8B! d \Rightarrow \left( P \right):2x + y + z - 1 = 0.\,\,\,\)

\(\Delta\) đi qua B và vuông góc với \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabgk % DiElabfs5aejabgkOimpaabmaabaGaamiuaaGaayjkaiaawMcaaaaa % !3EFA! d \Rightarrow \Delta \subset \left( P \right)\)

Gọi H, K lần lượt là hình chiếu của A lên (P) và ta có \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaadI % eacqGHKjYOcaWGbbGaam4saaaa!3AD2! AH \le AK\)

Do đó để khoảng cách từ A đến \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taam % isaiabgIGiolabfs5aebaa!3C08! \Rightarrow H \in \Delta \) là nhỏ nhất \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taam % isaiabgIGiolabfs5aebaa!3C08! \Rightarrow H \in \Delta \)

Phương trình AH đi qua A và nhận \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8Haaeaaca % WG1bWaaSbaaSqaaiaadsgaaeqaaaGccaGLxdcacqGH9aqpdaqadaqa % aiaaikdacaGG7aGaaGymaiaacUdacaaIXaaacaGLOaGaayzkaaaaaa!4000! \overrightarrow {{u_d}} = \left( {2;1;1} \right)\) là 1 VTCP là \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabe % qaaiaadIhacqGH9aqpcaaI2aGaey4kaSIaaGOmaiaadshaaeaacaWG % 5bGaeyypa0JaaG4maiabgUcaRiaadshaaeaacaWG6bGaeyypa0Jaey % OeI0IaaGOmaiabgUcaRiaadshaaaGaay5Eaaaaaa!4695! \left\{ \begin{array}{l} x = 6 + 2t\\ y = 3 + t\\ z = - 2 + t \end{array} \right.\)

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGib % GaeyicI4SaamyqaiaadIeacqGHshI3caWGibWaaeWaaeaacaaI2aGa % ey4kaSIaaGOmaiaadshacaGG7aGaaG4maiabgUcaRiaadshacaGG7a % GaeyOeI0IaaGOmaiabgUcaRiaadshaaiaawIcacaGLPaaaaeaacaWG % ibGaeyicI48aaeWaaeaacaWGqbaacaGLOaGaayzkaaGaeyO0H4TaaG % OmamaabmaabaGaaGOnaiabgUcaRiaaikdacaWG0baacaGLOaGaayzk % aaGaey4kaSIaaG4maiabgUcaRiaadshacqGHsislcaaIYaGaey4kaS % IaamiDaiabgkHiTiaaigdacqGH9aqpcaaIWaGaeyi1HSTaaGOnaiaa % dshacqGHRaWkcaaIXaGaaGOmaiabg2da9iaaicdacqGHuhY2caWG0b % Gaeyypa0JaeyOeI0IaaGOmaaqaaiabgkDiElaadIeadaqadaqaaiaa % ikdacaGG7aGaaGymaiaacUdacqGHsislcaaI0aaacaGLOaGaayzkaa % aaaaa!7817! \begin{array}{l} H \in AH \Rightarrow H\left( {6 + 2t;3 + t; - 2 + t} \right)\\ H \in \left( P \right) \Rightarrow 2\left( {6 + 2t} \right) + 3 + t - 2 + t - 1 = 0 \Leftrightarrow 6t + 12 = 0 \Leftrightarrow t = - 2\\ \Rightarrow H\left( {2;1; - 4} \right) \end{array}\)

đi qua B, H nhận \(% MathType!MTEF!2!1!+- % feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8Haaeaaca % WGcbGaamisaaGaay51GaWaaeWaaeaacaaIXaGaai4oaiaaigdacaGG % 7aGaeyOeI0IaaG4maaGaayjkaiaawMcaaaaa!3F63! \overrightarrow {BH} \left( {1;1; - 3} \right)\) là 1 VTCP.

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