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Giải bài tập 5.49 trang 63 SGK Toán 12 tập 2 - Kết nối tri thức

Trong không gian Oxyz, tính góc giữa mặt phẳng (P): \(x + y + z - 1 = 0\) và mặt phẳng Oxy.

Tổꦿng hợp đề thi học kì 2 lớp 12 tất cả các môn - Kết nối tri thức

Toán - Văn - Anh - Hoá - Sinh - Sử - Địa
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Đề bài

Trong không gian Oxyz, tính góc giữa mặt phẳng (P): \(x + y + z - 1 = 0\) và mặt phẳng Oxy.

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về góc giữa hai mặt phẳng để tính: Trong không gian Oxyz, cho hai mặt phẳng (P), (Q) tương ứng có các vectơ pháp tuyến là \(\overrightarrow n  = \left( {A;B;C} \right),\overrightarrow {n'}  = \left( {A';B';C'} \right)\). Khi đó, góc giữa (P) và (Q), kí hiệu là ((P), (Q)) được tính theo công thức:

\(\cos \left( {\left( P \right),\left( Q \right)} \right) = \left| {\cos \left( {\overrightarrow n ,\overrightarrow {n'} } \right)} \right| = \frac{{\left| {AA' + BB' + CC'} \right|}}{{\sqrt {{A^2} + {B^2} + {C^2}} .\sqrt {A{'^2} + B{'^2} + C{'^2}} }}\).

Lời giải chi tiết

Mặt phẳng (P) nhận \(\overrightarrow {{n_1}} \left( {1;1;1} \right)\) làm một vectơ pháp tuyến. Mặt phẳng (Oxy) nhận \(\overrightarrow k \left( {0;0;1} \right)\) làm một vectơ pháp tuyến. Ta có: \(\cos \left( {\left( P \right),\left( {Oxy} \right)} \right) = \frac{{\left| {1.0 + 1.0 + 1.1} \right|}}{{\sqrt {{1^2} + {1^2} + {1^2}} .\sqrt {{0^2} + {0^2} + {1^2}} }} = \frac{1}{{\sqrt 3 }} \Rightarrow \left( {\left( P \right),\left( {Oxy} \right)} \right) \approx 54,{7^o}\)

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