Hướng dẫn giải Bài 1 (Trang 91 SGK Toán Hình học 12)

1. Cho bốn điểm A(1; 0; 0), B(0; 1; 0), C(0; 0; 1), D(-2; 1; -1). a) Chứng minh A, B, C, D là bốn đỉnh của một tứ diện. b) Tìm góc giữa hai đường thẳng AB và CD. c) Tính độ dài đường cao của hình chóp A.BCD. Giải: a) Đường thẳng AB đi qua <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mfenced><mrow><mn>1</mn><mo>;</mo><mo> </mo><mn>0</mn><mo>;</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> có vectơ chỉ phương <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover></math>=<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>1</mn><mo>;</mo><mo> </mo><mn>0</mn><mo>;</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>. Đường thẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>D</mi></math> đi qua <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mfenced><mrow><mn>0</mn><mo>;</mo><mo> </mo><mn>0</mn><mo>;</mo><mo> </mo><mn>1</mn></mrow></mfenced></math> có vectơ chỉ phương <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>C</mi><mi>D</mi></mrow><mo>→</mo></mover></math> = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>2</mn><mo>;</mo><mo> </mo><mn>1</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>2</mn></mrow></mfenced></math> Ta có <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>C</mi></mrow><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mo>-</mo><mn>1</mn><mo>;</mo><mo> </mo><mn>0</mn><mo>;</mo><mo> </mo><mn>1</mn></mrow></mfenced></math> <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>2</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>;</mo><mo> </mo><mn>1</mn></mrow></mfenced><mo> </mo><mo>⇒</mo><mo> </mo><mover><mi>n</mi><mo>→</mo></mover><mo>.</mo><mover><mrow><mi>A</mi><mi>C</mi></mrow><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mo>.</mo><mn>0</mn><mo>+</mo><mn>1</mn><mo>.</mo><mn>1</mn><mo>=</mo><mn>3</mn><mo>≠</mo><mn>0</mn><mo>.</mo></math> Suy ra <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>D</mi></math> chéo nhau nên <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mo> </mo><mi>B</mi><mo>,</mo><mo> </mo><mi>C</mi><mo>,</mo><mo> </mo><mi>D</mi></math> là bốn đỉnh của một tứ diện. Cách khác: 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><mn>0</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>;</mo><mo> </mo><mn>1</mn></mrow></mfenced></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mo>-</mo><mn>2</mn><mo>;</mo><mo> </mo><mn>0</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>1</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>1</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>2</mn></mrow></mfenced></math> Phương trình mp<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>B</mi><mi>C</mi><mi>D</mi></mrow></mfenced></math> là: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>2</mn><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>0</mn><mo>⇔</mo><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>-</mo><mn>2</mn><mi>z</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mn>1</mn></mfenced></math> Tọa độ điểm <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> không thỏa <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mn>1</mn></mfenced></math> nên <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>∉</mo><mi>m</mi><mi>p</mi><mfenced><mrow><mi>B</mi><mi>C</mi><mi>D</mi></mrow></mfenced></math> Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mo> </mo><mi>B</mi><mo>,</mo><mo> </mo><mi>C</mi><mo>,</mo><mo> </mo><mi>D</mi></math> là bốn đỉnh của một tứ diện. b) 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><mo>-</mo><mn>1</mn><mo>;</mo><mo> </mo><mn>1</mn><mo>;</mo><mo> </mo><mn>0</mn></mrow></mfenced><mo>;</mo><mo> </mo><mover><mrow><mi>C</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mo>-</mo><mn>2</mn><mo>;</mo><mo> </mo><mn>1</mn><mo>;</mo><mo> </mo><mo>-</mo><mn>2</mn></mrow></mfenced><mspace linebreak="newline"/><mi>cos</mi><mfenced><mrow><mi>A</mi><mi>B</mi><mo>,</mo><mo> </mo><mi>C</mi><mi>D</mi></mrow></mfenced><mo>=</mo><mfrac><mfenced open="|" close="|"><mrow><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>.</mo><mover><mrow><mi>C</mi><mi>D</mi></mrow><mo>→</mo></mover></mrow></mfenced><mrow><mi>A</mi><mi>B</mi><mo>.</mo><mi>C</mi><mi>D</mi></mrow></mfrac><mo>=</mo><mfrac><mfenced open="|" close="|"><mrow><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>0</mn></mrow></mfenced><mrow><msqrt><mn>2</mn></msqrt><mo>.</mo><msqrt><mn>9</mn></msqrt></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></math> Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>A</mi><mi>B</mi><mo>,</mo><mo> </mo><mi>C</mi><mi>D</mi></mrow></mfenced><mo>=</mo><mn>45</mn><mo>°</mo></math>. c) Phương trình <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>p</mi><mfenced><mrow><mi>B</mi><mi>C</mi><mi>D</mi></mrow></mfenced></math> là: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>-</mo><mn>2</mn><mi>z</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></math> Độ dài đường cao của hình chóp <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>.</mo><mi>B</mi><mi>C</mi><mi>D</mi></math> là khoảng cách từ <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> đến <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>p</mi><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>1</mn><mo>+</mo><mn>2</mn></mrow></mfenced><msqrt><mn>1</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>4</mn></msqrt></mfrac><mo>=</mo><mn>1</mn><mo>.</mo></math>