Bài 2. Hai đường thẳng vuông góc
Hướng dẫn giải Hoạt động 2 (Trang 94 SGK Toán Hình học 11)
<p><strong class="content_question">Đề bài</strong></p>
<p>Cho hình lập phương <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mi>C</mi><mi>D</mi><mo>.</mo><mi>A</mi><mo>'</mo><mi>B</mi><mo>'</mo><mi>C</mi><mo>'</mo><mi>D</mi><mo>'</mo></math></p>
<p>a) Hãy phân tích các vecto <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>C</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>;</mo><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover></math> theo ba vecto <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>;</mo><mo> </mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>;</mo><mover><mrow><mi>A</mi><mi>A</mi><mo>'</mo></mrow><mo>→</mo></mover></math></p>
<p>b) Tính <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mrow><mover><mrow><mi>A</mi><mi>C</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>,</mo><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover></mrow></mfenced></math> và từ đó suy ra <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>C</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>;</mo><mo> </mo><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover></math> vuông góc với nhau</p>
<p><strong class="content_detail">Lời giải chi tiết</strong></p>
<p><img src="https://img.loigiaihay.com/picture/2018/0917/cau-2-trang-94.PNG" alt="" width="313" height="258" /></p>
<p>a) </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>C</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>=</mo><mover><mrow><mi>A</mi><mi>C</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>A</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>=</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>A</mi><mo>'</mo></mrow><mo>→</mo></mover><mspace linebreak="newline"/><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>=</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>-</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover></math></p>
<p>b)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mrow><mover><mrow><mi>A</mi><mi>C</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>,</mo><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover></mrow></mfenced><mo>=</mo><mfrac><mrow><mover><mrow><mi>A</mi><mi>C</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>.</mo><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover></mrow><mrow><mfenced open="|" close="|"><mover><mrow><mi>A</mi><mi>C</mi><mo>'</mo></mrow><mo>→</mo></mover></mfenced><mo>.</mo><mfenced open="|" close="|"><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover></mfenced></mrow></mfrac><mspace linebreak="newline"/><mover><mrow><mi>A</mi><mi>C</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>.</mo><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>A</mi><mo>'</mo></mrow><mo>→</mo></mover></mrow></mfenced><mo>.</mo><mfenced><mrow><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>-</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover></mrow></mfenced><mspace linebreak="newline"/><mo>=</mo><mfenced><mrow><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>A</mi><mo>'</mo></mrow><mo>→</mo></mover></mrow></mfenced><mo>.</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>-</mo><mo>(</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>A</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>)</mo><mo>.</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mspace linebreak="newline"/><mo>=</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>.</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>.</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>A</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>.</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>-</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>.</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>-</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>.</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo>-</mo><mover><mrow><mi>A</mi><mi>A</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>.</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mo> </mo><mo> </mo><mo>(</mo><mn>1</mn><mo>)</mo></math></p>
<p><span id="MathJax-Element-6-Frame" class="mjx-chtml MathJax_CHTML" style="margin: 0px; padding: 1px 0px; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 21.78px; letter-spacing: normal; overflow-wrap: 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"><mtable columnalign="right left" rowspacing=".5em" columnspacing="thickmathspace" displaystyle="true"><mtr><mtd /><mtd><mi>a</mi><mo stretchy="false">)</mo><mspace width="thinmathspace" /></mtd></mtr><mtr><mtd /><mtd><mover><mrow><mi>A</mi><msup><mi>C</mi><mo>&#x2032;</mo></msup></mrow><mo>&#x2192;</mo></mover><mo>=</mo><mover><mrow><mi>A</mi><mi>C</mi></mrow><mo>&#x2192;</mo></mover><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>+</mo></mrow></mrow><mover><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">A</mi><mi mathvariant="normal">A</mi></mrow></mrow><mo>&#x2192;</mo></mover><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi></mi><mo>&#x2032;</mo></msup></mrow></mrow><mspace width="thinmathspace" /><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo>=</mo></mrow></mrow><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>&#x2192;</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>&#x2192;</mo></mover><mo>+</mo><mover><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">A</mi><mi mathvariant="normal">A</mi></mrow></mrow><mo>&#x2192;</mo></mover><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><msup><mi></mi><mo>&#x2032;</mo></msup></mrow></mrow></mtd></mtr><mtr><mtd /><mtd><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>&#x2192;</mo></mover><mo>=</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>&#x2192;</mo></mover><mo>&#x2212;</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>&#x2192;</mo></mover></mtd></mtr><mtr><mtd /><mtd><mi>b</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd /><mtd><mi>cos</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mover><mrow><mi>A</mi><msup><mi>C</mi><mo>&#x2032;</mo></msup></mrow><mo>&#x2192;</mo></mover><mo>,</mo><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>&#x2192;</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow class="MJX-TeXAtom-ORD"><mover><mrow><mi>A</mi><msup><mi>C</mi><mo>&#x2032;</mo></msup></mrow><mo>&#x2192;</mo></mover><mo>.</mo><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>&#x2192;</mo></mover></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mover><mrow><mi>A</mi><msup><mi>C</mi><mo>&#x2032;</mo></msup></mrow><mo>&#x2192;</mo></mover><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mo>.</mo><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>&#x2192;</mo></mover><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd /><mtd><mover><mrow><mi>A</mi><msup><mi>C</mi><mo>&#x2032;</mo></msup></mrow><mo>&#x2192;</mo></mover><mo>.</mo><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>&#x2192;</mo></mover><mo>=</mo><mo stretchy="false">(</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>&#x2192;</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>&#x2192;</mo></mover><mo>+</mo><mover><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">A</mi><msup><mi mathvariant="normal">A</mi><mo>&#x2032;</mo></msup></mrow></mrow><mo>&#x2192;</mo></mover><mo stretchy="false">)</mo><mo>.</mo><mo stretchy="false">(</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>&#x2192;</mo></mover><mo>&#x2212;</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>&#x2192;</mo></mover><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd /><mtd><mo>=</mo><mo stretchy="false">(</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>&#x2192;</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>&#x2192;</mo></mover><mo>+</mo><mover><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">A</mi><msup><mi mathvariant="normal">A</mi><mo>&#x2032;</mo></msup></mrow></mrow><mo>&#x2192;</mo></mover><mo stretchy="false">)</mo><mo>.</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>&#x2192;</mo></mover><mo>&#x2212;</mo><mo stretchy="false">(</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>&#x2192;</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>&#x2192;</mo></mover><mo>+</mo><mover><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">A</mi><msup><mi mathvariant="normal">A</mi><mo>&#x2032;</mo></msup></mrow></mrow><mo>&#x2192;</mo></mover><mo stretchy="false">)</mo><mo>.</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>&#x2192;</mo></mover></mtd></mtr><mtr><mtd /><mtd><mo>=</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>&#x2192;</mo></mover><mo>.</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>&#x2192;</mo></mover><mo>+</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>&#x2192;</mo></mover><mo>.</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>&#x2192;</mo></mover><mo>+</mo><mover><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">A</mi><msup><mi mathvariant="normal">A</mi><mo>&#x2032;</mo></msup></mrow></mrow><mo>&#x2192;</mo></mover><mo>.</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>&#x2192;</mo></mover><mo>&#x2212;</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>&#x2192;</mo></mover><mo>.</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>&#x2192;</mo></mover><mo>&#x2212;</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>&#x2192;</mo></mover><mo>.</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>&#x2192;</mo></mover><mo>&#x2212;</mo><mover><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">A</mi><msup><mi mathvariant="normal">A</mi><mo>&#x2032;</mo></msup></mrow></mrow><mo>&#x2192;</mo></mover><mo>.</mo><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>&#x2192;</mo></mover><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></math>"><span class="MJX_Assistive_MathML" role="presentation"><br /><mtr><mtd></mtd><mtd><br /></mtd></mtr><mtr><mtd></mtd><mtd><br /></mtd></mtr><mtr><mtd></mtd><mtd><br /></mtd></mtr><mtr><mtd></mtd><mtd><br /></mtd></mtr><mtr><mtd></mtd><mtd><br /></mtd></mtr><mtr><mtd></mtd><mtd><br /></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false"></mo></mtd></mtr></span></span>Hình lập phương <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mi>C</mi><mi>D</mi><mo>.</mo><mi>A</mi><mo>'</mo><mi>B</mi><mo>'</mo><mi>C</mi><mo>'</mo><mi>D</mi><mo>'</mo></math> nên <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mo>,</mo><mi>A</mi><mi>D</mi><mo>,</mo><mi>A</mi><mi>A</mi><mo>'</mo></math> đôi một vuông góc với nhau</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mn>1</mn></mfenced><mo>=</mo><mover><mn>0</mn><mo>→</mo></mover><mo>+</mo><msup><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mn>2</mn></msup><mo>+</mo><mover><mn>0</mn><mo>→</mo></mover><mo>-</mo><msup><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>→</mo></mover><mn>2</mn></msup><mo>-</mo><mover><mn>0</mn><mo>→</mo></mover><mo>-</mo><mover><mn>0</mn><mo>→</mo></mover><mo>=</mo><mn>0</mn><mo> </mo><mfenced><mrow><mi>A</mi><mi>B</mi><mo>=</mo><mi>A</mi><mi>D</mi></mrow></mfenced><mspace linebreak="newline"/><mo>⇒</mo><mi>cos</mi><mfenced><mrow><mover><mrow><mi>A</mi><mi>C</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>,</mo><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover></mrow></mfenced><mo>=</mo><mfrac><mrow><mover><mrow><mi>A</mi><mi>C</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>.</mo><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover></mrow><mrow><mfenced open="|" close="|"><mover><mrow><mi>A</mi><mi>C</mi><mo>'</mo></mrow><mo>→</mo></mover></mfenced><mo>.</mo><mfenced open="|" close="|"><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover></mfenced></mrow></mfrac><mo>=</mo><mn>0</mn><mspace linebreak="newline"/><mo>⇒</mo><mfenced><mrow><mover><mrow><mi>A</mi><mi>C</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>,</mo><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover></mrow></mfenced><mo>=</mo><mn>90</mn><mo>°</mo></math></p>
<p>Vậy hai vecto trên vuông góc với nhau.</p>
Xem lời giải bài tập khác cùng bài
Lý thuyết hai đường thẳng vuông góc
Xem lời giải
Hướng dẫn giải Hoạt động 1 (Trang 93 SGK Toán Hình học 11)
Xem lời giải
Hướng dẫn giải Hoạt động 3 (Trang 95 SGK Toán Hình học 11)
Xem lời giải
Hướng dẫn giải Hoạt động 4 (Trang 97 SGK Toán Hình học 11)
Xem lời giải
Hướng dẫn giải Hoạt động 5 (Trang 97 SGK Toán Hình học 11)
Xem lời giải
Hướng dẫn giải Bài 1 (Trang 97 SGK Toán Hình học 11)
Xem lời giải
Hướng dẫn giải Bài 2 (Trang 97 SGK Toán Hình học 11)
Xem lời giải
Hướng dẫn giải Bài 3 (Trang 97 SGK Toán Hình học 11)
Xem lời giải
Hướng dẫn giải Bài 4 (Trang 98 SGK Toán Hình học 11)
Xem lời giải
Hướng dẫn giải Bài 5 (Trang 98 SGK Toán Hình học 11)
Xem lời giải
Hướng dẫn giải Bài 6 (Trang 98 SGK Toán Hình học 11)
Xem lời giải
Hướng dẫn giải Bài 7 (Trang 98 SGK Toán Hình học 11)
Xem lời giải
Hướng dẫn giải Bài 8 (Trang 98 SGK Toán Hình học 11)
Xem lời giải