Hướng dẫn giải Bài 11 (Trang 93 SGK Toán Hình học 12)

11. Viết phương trình đường thẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo></math> vuông góc với mặt phẳng tọa độ <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>O</mi><mi>x</mi><mi>z</mi></mrow></mfenced></math> và cắt hai đường thẳng: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo> </mo><mo>:</mo><mo> </mo><mfenced open="{" close=""><mtable><mtr><mtd><mi>x</mi><mo>=</mo><mi>t</mi></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mo>-</mo><mn>4</mn><mo>+</mo><mi>t</mi></mtd></mtr><mtr><mtd><mi>z</mi><mo>=</mo><mn>3</mn><mo>-</mo><mi>t</mi></mtd></mtr></mtable></mfenced><mo>;</mo><mo> </mo><mo> </mo><mo> </mo><mi>d</mi><mo>'</mo><mo> </mo><mo>:</mo><mo> </mo><mfenced open="{" close=""><mtable><mtr><mtd><mi>x</mi><mo>=</mo><mn>1</mn><mo>-</mo><mn>2</mn><mi>t</mi><mo>'</mo></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mo>-</mo><mn>3</mn><mo>+</mo><mi>t</mi><mo>'</mo></mtd></mtr><mtr><mtd><mi>z</mi><mo>=</mo><mn>4</mn><mo>-</mo><mn>5</mn><mi>t</mi><mo>'</mo></mtd></mtr></mtable></mfenced></math> Giải: <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo></math> vuông góc với mặt phẳng tọa độ <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>O</mi><mi>x</mi><mi>y</mi><mi>z</mi></mrow></mfenced></math> nên <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo></math> có vecto chỉ phương là <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>j</mi><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mn>0</mn><mo>;</mo><mo> </mo><mn>1</mn><mo>;</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>. Gọi <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mfenced><mrow><mi>t</mi><mo>;</mo><mo> </mo><mo>-</mo><mn>4</mn><mo>+</mo><mi>t</mi><mo>;</mo><mo> </mo><mn>3</mn><mo>-</mo><mi>t</mi></mrow></mfenced></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>'</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mn>2</mn><mi>t</mi><mo>'</mo><mo>;</mo><mo> </mo><mo>-</mo><mn>3</mn><mo>+</mo><mi>t</mi><mo>'</mo><mo>;</mo><mo> </mo><mn>4</mn><mo>-</mo><mn>5</mn><mi>t</mi><mo>'</mo></mrow></mfenced></math> lần lượt là giao điểm của <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo></math> với <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>'</mo><mfenced><mrow><mi>h</mi><mo>.</mo><mn>34</mn></mrow></mfenced></math>, ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>M</mi><mi>M</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>=</mo><mi>k</mi><mover><mi>j</mi><mo>→</mo></mover></math>. Suy ra: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable><mtr><mtd><mn>1</mn><mo>-</mo><mn>2</mn><mi>t</mi><mo>'</mo><mo>-</mo><mi>t</mi><mo>=</mo><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mn>1</mn></mfenced></mtd></mtr><mtr><mtd><mn>1</mn><mo>+</mo><mi>t</mi><mo>'</mo><mo>-</mo><mi>t</mi><mo>=</mo><mi>k</mi><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mn>2</mn></mfenced></mtd></mtr><mtr><mtd><mn>1</mn><mo>-</mo><mn>5</mn><mi>t</mi><mo>'</mo><mo>+</mo><mi>t</mi><mo>=</mo><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mn>3</mn></mfenced><mo> </mo><mo> </mo><mo> </mo><mo> </mo></mtd></mtr></mtable></mfenced></math>                      <img class="wscnph" src="https://static.colearn.vn:8413/v1.0/upload/library/20022022/31eed8bf-92b9-4c9d-8f39-09a5e85c190a.PNG" /> Từ <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mn>1</mn></mfenced></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mn>3</mn></mfenced></math> suy ra <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>t</mi><mo>=</mo><mfrac><mn>3</mn><mn>7</mn></mfrac></mtd></mtr><mtr><mtd><mi>t</mi><mo>'</mo><mo>=</mo><mfrac><mn>2</mn><mn>7</mn></mfrac></mtd></mtr></mtable></mfenced></math> Thay <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mn>3</mn><mn>7</mn></mfrac></math> vào tọa độ <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> ta được <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mfenced><mrow><mfrac><mn>3</mn><mn>7</mn></mfrac><mo>;</mo><mo> </mo><mfrac><mrow><mo>-</mo><mn>25</mn></mrow><mn>7</mn></mfrac><mo>;</mo><mo> </mo><mfrac><mn>18</mn><mn>7</mn></mfrac></mrow></mfenced></math> Vậy phương trình tham số của đường thẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo></math> là: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mn>3</mn><mn>7</mn></mfrac></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>25</mn></mrow><mn>7</mn></mfrac><mo>+</mo><mi>t</mi></mtd></mtr><mtr><mtd><mi>z</mi><mo>=</mo><mfrac><mn>18</mn><mn>7</mn></mfrac></mtd></mtr></mtable></mfenced></math>