Hướng dẫn giải Bài 7 (Trang 100 SGK Toán Hình học 12)

Trong không gian cho hai đường thẳng d1 và d2 có phương trình <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>d</mi><mn>1</mn></msub><mo> </mo><mo>:</mo><mfenced open="{" close=""><mrow><mtable><mtr><mtd><mi>x</mi><mo>=</mo><mn>1</mn><mo>-</mo><mi>t</mi></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mi>t</mi></mtd></mtr><mtr><mtd><mi>z</mi><mo>=</mo><mo>-</mo><mi>t</mi></mtd></mtr></mtable><mo> </mo></mrow></mfenced><mo> </mo><mo> </mo><mo> </mo><msub><mi>d</mi><mrow><mn>2</mn><mo> </mo></mrow></msub><mo>:</mo><mfenced open="{" close=""><mrow><mtable><mtr><mtd><mi>x</mi><mo>=</mo><mn>2</mn><mi>t</mi><mo>'</mo></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>+</mo><mi>t</mi><mo>'</mo></mtd></mtr><mtr><mtd><mi>z</mi><mo>=</mo><mi>t</mi><mo>'</mo></mtd></mtr></mtable><mo> </mo></mrow></mfenced></math> a) Chứng minh rằng hai đường thẳng d1 và d2 chéo nhau. b) Viết phương trình của mặt phẳng <span id="MathJax-Element-2-Frame" class="mjx-chtml MathJax_CHTML" style="box-sizing: border-box; margin: 0px; padding: 1px 0px; overflow-wrap: normal; word-break: break-word; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 18.08px; letter-spacing: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>&#x03B1;</mi><mo stretchy="false">)</mo></math>">(α)<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></math> chứa d1 và song song với d2. Giải: a) Đường thẳng d1 đi qua M1(1; 0; 0) vecto chỉ phương là <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>u</mi><mn>1</mn></msub><mo>→</mo></mover><mo> </mo><mo>=</mo><mo>(</mo><mo>-</mo><mn>1</mn><mo>;</mo><mn>1</mn><mo>;</mo><mo>-</mo><mn>1</mn><mo>)</mo></math> Đường thẳng d2 đi qua M2(0; -1; 0) vecto chỉ phương là <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>u</mi><mn>2</mn></msub><mo>→</mo></mover><mo> </mo><mo>=</mo><mo>(</mo><mn>2</mn><mo>;</mo><mn>1</mn><mo>;</mo><mn>1</mn><mo>)</mo></math> Ta có <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>n</mi><mo>→</mo></mover><mo> </mo><mo>=</mo><mfenced open="[" close="]"><mrow><mover><msub><mi>u</mi><mn>1</mn></msub><mo>→</mo></mover><mo> </mo><mo>.</mo><mover><msub><mi>u</mi><mn>2</mn></msub><mo>→</mo></mover></mrow></mfenced><mo>=</mo><mo>(</mo><mn>2</mn><mo>;</mo><mo>-</mo><mn>1</mn><mo>;</mo><mo>-</mo><mn>3</mn><mo>)</mo></math>  <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><msub><mi>M</mi><mn>1</mn></msub><msub><mi>M</mi><mn>2</mn></msub></mrow><mo>→</mo></mover><mo>=</mo><mo>(</mo><mo>-</mo><mn>1</mn><mo>;</mo><mo>-</mo><mn>1</mn><mo>;</mo><mn>0</mn><mo>)</mo></math> Suy ra <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>n</mi><mo>→</mo></mover><mo>.</mo><mover><mrow><msub><mi>M</mi><mn>1</mn></msub><msub><mi>M</mi><mn>2</mn></msub><mo> </mo></mrow><mo>→</mo></mover><mo>=</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>=</mo><mo>-</mo><mn>1</mn></math>≠0 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mi>ậ</mi><mi>y</mi><mo> </mo><msub><mi>d</mi><mn>1</mn></msub><mo> </mo><mi>v</mi><mi>à</mi><mo> </mo><msub><mi>d</mi><mn>2</mn></msub><mo> </mo><mi>c</mi><mi>h</mi><mi>é</mi><mi>o</mi><mo> </mo><mi>n</mi><mi>h</mi><mi>a</mi><mi>u</mi><mo>.</mo></math> b) Vecto pháp tuyến mp<span id="MathJax-Element-15-Frame" class="mjx-chtml MathJax_CHTML" style="box-sizing: border-box; margin: 0px; padding: 1px 0px; overflow-wrap: normal; word-break: break-word; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: 400; font-size: 18.08px; letter-spacing: normal; word-spacing: 0px; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; color: #131313; font-family: Quicksand; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; background-color: #ffffff; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>&#x03B1;</mi><mo stretchy="false">)</mo></math>"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></math> là: <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>n</mi><mo>→</mo></mover><mo> </mo><mo>=</mo><mfenced open="[" close="]"><mrow><mover><msub><mi>u</mi><mn>1</mn></msub><mo>→</mo></mover><mo> </mo><mo>.</mo><mover><msub><mi>u</mi><mn>2</mn></msub><mo>→</mo></mover></mrow></mfenced><mo>=</mo><mo>(</mo><mn>2</mn><mo>;</mo><mo>-</mo><mn>1</mn><mo>;</mo><mo>-</mo><mn>3</mn><mo>)</mo></math>  Phương trình mp<span id="MathJax-Element-17-Frame" class="mjx-chtml MathJax_CHTML" style="box-sizing: border-box; margin: 0px; padding: 1px 0px; overflow-wrap: normal; word-break: break-word; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: 400; font-size: 18.08px; letter-spacing: normal; word-spacing: 0px; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; color: #131313; font-family: Quicksand; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; background-color: #ffffff; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>&#x03B1;</mi><mo stretchy="false">)</mo></math>">(α)<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></math> là: <span id="MathJax-Element-18-Frame" class="mjx-chtml MathJax_CHTML" style="box-sizing: border-box; margin: 0px; padding: 1px 0px; overflow-wrap: normal; word-break: break-word; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: 400; font-size: 18.08px; letter-spacing: normal; word-spacing: 0px; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; color: #131313; font-family: Quicksand; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; widows: 2; -webkit-text-stroke-width: 0px; background-color: #ffffff; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial; position: relative;" tabindex="0" role="presentation" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo stretchy="false">(</mo><mi>x</mi><mo>&#x2212;</mo><mn>1</mn><mo stretchy="false">)</mo><mo>&#x2212;</mo><mi>y</mi><mo>&#x2212;</mo><mn>3</mn><mi>z</mi><mo>=</mo><mn>0</mn></math>"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>(</mo><mo> </mo><mi>x</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>−</mo><mi>y</mi><mo>−</mo><mn>3</mn><mi>z</mi><mo>=</mo><mn>0</mn><mo> </mo><mi>h</mi><mi>a</mi><mi>y</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>−</mo><mn>3</mn><mi>z</mi><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></math>

Hướng dẫn Giải Bài 7 (trang 100, SGK Toán 12, Hình học)

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