Hướng dẫn giải Bài 3 (Trang 92 SGK Toán Hình học 12)

3. Cho bốn điểm <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mfenced><mrow><mo>-</mo><mn>2</mn><mo>;</mo><mo> </mo><mn>6</mn><mo>;</mo><mo> </mo><mn>3</mn></mrow></mfenced><mo>,</mo><mo> </mo><mi>B</mi><mfenced><mrow><mn>1</mn><mo>;</mo><mo> </mo><mn>0</mn><mo>;</mo><mo> </mo><mn>6</mn></mrow></mfenced><mo>,</mo><mo> </mo><mi>C</mi><mfenced><mrow><mn>0</mn><mo>;</mo><mo> </mo><mn>2</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced><mo>,</mo><mo> </mo><mi>D</mi><mfenced><mrow><mn>1</mn><mo>;</mo><mo> </mo><mn>4</mn><mo>;</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>. a) Viết phương trình mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>B</mi><mi>C</mi><mi>D</mi></mrow></mfenced></math>. Suy ra <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mi>C</mi><mi>D</mi></math> là một tứ diện. b) Tính chiều cao <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>H</mi></math> của tứ diện <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mi>C</mi><mi>D</mi></math>. c) Viết phương trình mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math> chứa <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi></math> và song song với <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>D</mi></math>. Giải: a) Ta có <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>B</mi><mi>C</mi></mrow><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mo>-</mo><mn>1</mn><mo>;</mo><mo> </mo><mn>2</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>7</mn></mrow></mfenced><mo>,</mo><mo> </mo><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mn>0</mn><mo>;</mo><mo> </mo><mn>4</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>6</mn></mrow></mfenced></math> Mp<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>B</mi><mi>C</mi><mi>D</mi></mrow></mfenced></math> có vectơ pháp tuyến: <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>n</mi><mo>→</mo></mover><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mover><mrow><mi>B</mi><mi>C</mi></mrow><mo>→</mo></mover><mo>,</mo></mtd><mtd><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mrow><mn>16</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>6</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>4</mn></mrow></mfenced></math> Mp<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>B</mi><mi>C</mi><mi>D</mi></mrow></mfenced></math> đi qua <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mfenced><mrow><mn>1</mn><mo>;</mo><mo> </mo><mn>0</mn><mo>;</mo><mo> </mo><mn>6</mn></mrow></mfenced></math> và có vectơ pháp tuyến <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>n</mi><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mn>16</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>6</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>4</mn></mrow></mfenced></math>nên có phương trình là: <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>6</mn><mi>y</mi><mo>-</mo><mn>4</mn><mfenced><mrow><mi>z</mi><mo>-</mo><mn>6</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>⇔</mo><mn>8</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>-</mo><mn>2</mn><mi>z</mi><mo>+</mo><mn>4</mn><mo>=</mo><mn>0</mn></math> Vì <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>∉</mo><mfenced><mrow><mi>B</mi><mi>C</mi><mi>D</mi></mrow></mfenced></math> nên <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mi>C</mi><mi>D</mi></math> là một tứ diện. b) Chiều cao <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>H</mi></math> của tứ diện là khoảng cách từ <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> đến mp<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>B</mi><mi>C</mi><mi>D</mi></mrow></mfenced></math>. Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>H</mi><mo>=</mo><mi>d</mi><mfenced><mrow><mi>A</mi><mo>,</mo><mo> </mo><mfenced><mrow><mi>B</mi><mi>C</mi><mi>D</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mfrac><mfenced open="|" close="|"><mrow><mn>8</mn><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mo>-</mo><mn>3</mn><mo>.</mo><mn>6</mn><mo>-</mo><mn>2</mn><mo>.</mo><mn>3</mn><mo>+</mo><mn>4</mn></mrow></mfenced><msqrt><msup><mn>8</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></msqrt></mfrac><mo>=</mo><mfrac><mn>36</mn><msqrt><mn>77</mn></msqrt></mfrac></math> c) Ta có <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mn>3</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>6</mn><mo>;</mo><mo> </mo><mn>3</mn></mrow></mfenced><mo>,</mo><mo> </mo><mover><mrow><mi>C</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mn>1</mn><mo>;</mo><mo> </mo><mn>2</mn><mo>;</mo><mo> </mo><mn>1</mn></mrow></mfenced></math> Mp<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math> chứa <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover></math> và song song với <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>D</mi></math> có vectơ pháp tuyến <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>n</mi><mo>→</mo></mover><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>,</mo></mtd><mtd><mover><mrow><mi>C</mi><mi>D</mi></mrow><mo>→</mo></mover></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mrow><mo>-</mo><mn>12</mn><mo>;</mo><mo> </mo><mn>0</mn><mo>;</mo><mo> </mo><mn>12</mn></mrow></mfenced></math> Phương trình mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>α</mi></mfenced></math> là: <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>12</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mn>0</mn><mfenced><mrow><mi>y</mi><mo>-</mo><mn>0</mn></mrow></mfenced><mo>+</mo><mn>12</mn><mfenced><mrow><mi>z</mi><mo>-</mo><mn>6</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>⇔</mo><mi>x</mi><mo>-</mo><mi>z</mi><mo>+</mo><mn>5</mn><mo>=</mo><mn>0</mn><mo>.</mo></math>