Cách xét vị trí tương đối giữa hai mặt phẳng bằng phương pháp toạ độ - Toán 12

1. Cách xét vị trí tương đối giữa hai mặt phẳng bằng phương pháp toạ độ

Trong không gian Oxyz, cho hai mặt phẳng (P) và (Q) có phương trình: (P): Ax + By + Cz + D = 0, \({A^2} + {B^2} + {C^2} \ne 0\). (Q): A’x + B’y + C’z + D’ = 0, \(A{'^2} + B{'^2} + C{'^2} \ne 0\). Ta có (P) và (Q): - Song song khi \(\frac{A}{{A'}} = \frac{B}{{B'}} = \frac{C}{{C'}} \ne \frac{D}{{D'}}\). - Trùng nhau khi \(\frac{A}{{A'}} = \frac{B}{{B'}} = \frac{C}{{C'}} = \frac{D}{{D'}}\). - Cắt nhau khi \(\frac{A}{{A'}} \ne \frac{B}{{B'}}\) hoặc \(\frac{B}{{B'}} \ne \frac{C}{{C'}}\) hoặc \(\frac{C}{{C'}} \ne \frac{A}{{A'}}\). - Vuông góc khi \(AA' + BB' + CC' = 0\).

2. Ví dụ minh hoạ về xét vị trí tương đối giữa hai mặt phẳng bằng phương pháp toạ độ

Một công trường xây dựng nhà cao tầng đã thiết lập hệ toạ độ Oxyz. Hãy kiểm tra tính song song hoặc vuông góc giữa các mặt kính (P), (Q), (R) của một toà nhà, biết: (P): 3x + y – z + 2 = 0; (Q): 6x + 2y – 2z + 11 = 0; (R): x – 3y + 1 = 0.

Giải:

Các vecto pháp tuyến của các mặt phẳng \(\left( P \right)\), \(\left( Q \right)\), \(\left( R \right)\) lần lượt là \(\overrightarrow {{n_{\left( P \right)}}} = \left( {3;1; - 1} \right)\), \(\overrightarrow {{n_{\left( Q \right)}}} = \left( {6;2; - 2} \right)\) và \(\overrightarrow {{n_{\left( R \right)}}} = \left( {1; - 3;0} \right)\). Ta thấy rằng \(\frac{3}{6} = \frac{1}{2} = \frac{{ - 1}}{{ - 2}}\) nên \(\overrightarrow {{n_{\left( P \right)}}} \) và \(\overrightarrow {{n_{\left( Q \right)}}} \) là 2 vecto cùng phương. Từ đó suy ra \(\left( P \right)//\left( Q \right)\). Ta có \(\overrightarrow {{n_{\left( P \right)}}} .\overrightarrow {{n_{\left( R \right)}}} = 3.1 + 1.\left( { - 3} \right) + \left( { - 1} \right).0 = 0\) nên \(\overrightarrow {{n_{\left( P \right)}}} \) và \(\overrightarrow {{n_{\left( R \right)}}} \) có giá vuông góc với nhau. Suy ra \(\left( P \right) \bot \left( R \right)\). Ta có \(\overrightarrow {{n_{\left( Q \right)}}} .\overrightarrow {{n_{\left( R \right)}}} = 6.1 + 2.\left( { - 3} \right) + \left( { - 2} \right).0 = 0\) nên \(\overrightarrow {{n_{\left( Q \right)}}} \) và \(\overrightarrow {{n_{\left( R \right)}}} \) có giá vuông góc với nhau. Suy ra \(\left( Q \right) \bot \left( R \right)\).

3. Bài tập vận dụng

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