Lý thuyết Phương trình mặt phẳng

1. Vectơ pháp tuyến của mặt phẳng. - Cho mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>P</mi></mfenced></math> , vectơ <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>n</mi><mo>→</mo></mover><mo>≠</mo><mover><mn>0</mn><mo>→</mo></mover></math> mà giá của nó vuông góc với mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>P</mi></mfenced></math> thì <span id="MathJax-Element-4-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"><mover><mi>n</mi><mo>&#x2192;</mo></mover></math>"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>n</mi><mo>→</mo></mover></math> được gọi là vectơ pháp tuyến của mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>P</mi></mfenced></math>. *Cho mặt phẳng  <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>P</mi></mfenced></math>, cặp vectơ  <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>→</mo></mover><mo>≠</mo><mover><mn>0</mn><mo>→</mo></mover><mo>,</mo><mo> </mo><mover><mi>b</mi><mo>→</mo></mover><mo>≠</mo><mover><mn>0</mn><mo>→</mo></mover></math> không cùng phương mà  giá của chúng là hai đường thẳng song song hay nằm trong mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>P</mi></mfenced></math> được gọi là cặp vectơ chỉ phương của mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>P</mi></mfenced></math>. Khi đó vectơ <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><mi>a</mi><mo>→</mo></mover><mo>.</mo></mtd><mtd><mover><mi>b</mi><mo>→</mo></mover></mtd></mtr></mtable></mfenced></math> là vectơ pháp tuyến của mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>P</mi></mfenced></math>. *Nếu <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>a</mi><mo>→</mo></mover><mo>=</mo><mfenced><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>;</mo><mo> </mo><msub><mi>a</mi><mn>2</mn></msub><mo>;</mo><mo> </mo><msub><mi>a</mi><mn>3</mn></msub></mrow></mfenced><mo>,</mo><mo> </mo><mover><mi>b</mi><mo>→</mo></mover><mo>=</mo><mfenced><mrow><msub><mi>b</mi><mn>1</mn></msub><mo>;</mo><mo> </mo><msub><mi>b</mi><mn>2</mn></msub><mo>;</mo><mo> </mo><msub><mi>b</mi><mn>3</mn></msub></mrow></mfenced></math> thì : <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><mi>a</mi><mo>→</mo></mover><mo>.</mo></mtd><mtd><mover><mi>b</mi><mo>→</mo></mover></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mrow><mfenced open="|" close="|"><mtable><mtr><mtd><msub><mi>a</mi><mn>2</mn></msub></mtd><mtd><msub><mi>a</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd><msub><mi>b</mi><mn>2</mn></msub></mtd><mtd><msub><mi>b</mi><mn>3</mn></msub></mtd></mtr></mtable></mfenced><mo> </mo><mo>;</mo><mo> </mo><mfenced open="|" close="|"><mtable><mtr><mtd><msub><mi>a</mi><mn>3</mn></msub></mtd><mtd><msub><mi>a</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>b</mi><mn>3</mn></msub></mtd><mtd><msub><mi>b</mi><mn>1</mn></msub></mtd></mtr></mtable></mfenced><mo> </mo><mo>;</mo><mo> </mo><mfenced open="|" close="|"><mtable><mtr><mtd><msub><mi>a</mi><mn>1</mn></msub></mtd><mtd><msub><mi>a</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>b</mi><mn>1</mn></msub></mtd><mtd><msub><mi>b</mi><mn>2</mn></msub></mtd></mtr></mtable></mfenced></mrow></mfenced><mo>=</mo><mfenced><mrow><msub><mi>a</mi><mn>2</mn></msub><msub><mi>b</mi><mn>3</mn></msub><mo>-</mo><msub><mi>a</mi><mn>3</mn></msub><msub><mi>b</mi><mn>2</mn></msub><mo> </mo><mo>;</mo><mo> </mo><msub><mi>a</mi><mn>3</mn></msub><msub><mi>b</mi><mn>1</mn></msub><mo>-</mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>b</mi><mn>3</mn></msub><mo> </mo><mo>;</mo><mo> </mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>b</mi><mn>2</mn></msub><mo>-</mo><msub><mi>a</mi><mn>2</mn></msub><msub><mi>b</mi><mn>1</mn></msub><mo> </mo></mrow></mfenced><mo>.</mo></math> - Mặt phẳng hoàn toàn được xác định khi biết một điểm và một vectơ pháp tuyến của nó, hay một điểm thuộc mặt phẳng và cặp vectơ chỉ phương của nó. 2. Phương trình mặt phẳng. * Mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>P</mi></mfenced></math>  qua điểm <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mn>0</mn></msub><mfenced><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>;</mo><mo> </mo><msub><mi>y</mi><mn>0</mn></msub><mo>;</mo><mo> </mo><msub><mi>z</mi><mn>0</mn></msub></mrow></mfenced></math><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mspace width="thickmathspace"></mspace></math> và nhận <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>n</mi><mo>→</mo></mover><mfenced><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi></mrow></mfenced></math> làm vectơ pháp tuyến có phương trình có dạng:<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mfenced><mrow><mi>x</mi><mo>-</mo><msub><mi>x</mi><mn>0</mn></msub></mrow></mfenced><mo>+</mo><mi>B</mi><mfenced><mrow><mi>y</mi><mo>-</mo><msub><mi>y</mi><mn>0</mn></msub></mrow></mfenced><mo>+</mo><mi>C</mi><mfenced><mrow><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>0</mn></msub></mrow></mfenced><mo>=</mo><mn>0</mn></math> * Mọi mặt phẳng trong không gian có phương trình tổng quát có dạng:                             <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>x</mi><mo>+</mo><mi>B</mi><mi>y</mi><mo>+</mo><mi>C</mi><mi>z</mi><mo>+</mo><mi>D</mi><mo>=</mo><mn>0</mn><mo> </mo></math>ở đó <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><msup><mi>B</mi><mn>2</mn></msup><mo>+</mo><msup><mi>C</mi><mn>2</mn></msup><mo>></mo><mn>0</mn></math>                       Khi đó vectơ <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>n</mi><mo>→</mo></mover><mfenced><mrow><mi>A</mi><mo>;</mo><mi>B</mi><mo>;</mo><mi>C</mi></mrow></mfenced></math> là vectơ pháp tuyến của mặt phẳng. - Mặt phẳng đi qua ba điểm <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mfenced><mrow><mi>a</mi><mo>;</mo><mn>0</mn><mo>;</mo><mn>0</mn></mrow></mfenced><mo>,</mo><mo> </mo><mi>N</mi><mfenced><mrow><mn>0</mn><mo>;</mo><mi>b</mi><mo>;</mo><mn>0</mn></mrow></mfenced><mo>,</mo><mo> </mo><mi>C</mi><mfenced><mrow><mn>0</mn><mo>;</mo><mn>0</mn><mo>;</mo><mi>c</mi></mrow></mfenced></math> ở đó <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mi>c</mi><mo>≠</mo><mn>0</mn></math> có phương trình <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mi>a</mi></mfrac><mo>+</mo><mfrac><mi>y</mi><mi>b</mi></mfrac><mo>+</mo><mfrac><mi>z</mi><mi>c</mi></mfrac><mo>=</mo><mn>1</mn></math>. Phương trình này còn được gọi là phương trình mặt phẳng theo đoạn chắn. 3. Vị trí tương đối của hai mặt phẳng.  Cho hai mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>P</mi><mn>1</mn></msub></mfenced></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>P</mi><mn>2</mn></msub></mfenced></math> có phương trình : <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>P</mi><mn>1</mn></msub></mfenced><mo>:</mo><mo> </mo><msub><mi>A</mi><mn>1</mn></msub><mi>x</mi><mo>+</mo><msub><mi>B</mi><mn>1</mn></msub><mi>y</mi><mo>+</mo><msub><mi>C</mi><mn>1</mn></msub><mi>z</mi><mo>+</mo><msub><mi>D</mi><mn>1</mn></msub><mo>=</mo><mn>0</mn><mo>;</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>P</mi><mn>2</mn></msub></mfenced><mo>:</mo><mo> </mo><msub><mi>A</mi><mn>2</mn></msub><mi>x</mi><mo>+</mo><msub><mi>B</mi><mn>2</mn></msub><mi>y</mi><mo>+</mo><msub><mi>C</mi><mn>2</mn></msub><mi>z</mi><mo>+</mo><msub><mi>D</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn><mo>.</mo></math> Ta có <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>n</mi><mn>1</mn></msub><mo>→</mo></mover><mfenced><mrow><msub><mi>A</mi><mn>1</mn></msub><mo>;</mo><msub><mi>B</mi><mn>1</mn></msub><mo>;</mo><msub><mi>C</mi><mn>1</mn></msub></mrow></mfenced><mo>&perp;</mo><mfenced><msub><mi>P</mi><mn>1</mn></msub></mfenced></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>n</mi><mn>2</mn></msub><mo>→</mo></mover><mfenced><mrow><msub><mi>A</mi><mn>2</mn></msub><mo>;</mo><msub><mi>B</mi><mn>2</mn></msub><mo>;</mo><msub><mi>C</mi><mn>2</mn></msub></mrow></mfenced><mo>&perp;</mo><mfenced><msub><mi>P</mi><mn>2</mn></msub></mfenced></math>. Khi đó: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>P</mi><mn>1</mn></msub></mfenced><mo>&perp;</mo><mfenced><msub><mi>P</mi><mn>2</mn></msub></mfenced><mo>⇔</mo><mover><msub><mi>n</mi><mn>1</mn></msub><mo>→</mo></mover><mo>&perp;</mo><mover><msub><mi>n</mi><mn>2</mn></msub><mo>→</mo></mover><mo>⇔</mo><mover><msub><mi>n</mi><mn>1</mn></msub><mo>→</mo></mover><mo>.</mo><mover><msub><mi>n</mi><mn>2</mn></msub><mo>→</mo></mover><mo>⇔</mo><msub><mi>A</mi><mn>1</mn></msub><msub><mi>A</mi><mn>2</mn></msub><mo>+</mo><msub><mi>B</mi><mn>1</mn></msub><msub><mi>B</mi><mn>2</mn></msub><mo>+</mo><msub><mi>C</mi><mn>1</mn></msub><msub><mi>C</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>P</mi><mn>1</mn></msub></mfenced><mo>∥</mo><mfenced><msub><mi>P</mi><mn>2</mn></msub></mfenced><mo>⇔</mo><mover><msub><mi>n</mi><mn>1</mn></msub><mo>→</mo></mover><mo>=</mo><mi>k</mi><mo>.</mo><mover><msub><mi>n</mi><mn>2</mn></msub><mo>→</mo></mover></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mn>1</mn></msub><mo>≠</mo><mi>k</mi><mo>.</mo><msub><mi>D</mi><mn>2</mn></msub><mo> </mo><mfenced><mrow><mi>k</mi><mo>≠</mo><mn>0</mn></mrow></mfenced><mo>.</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>P</mi><mn>1</mn></msub></mfenced><mo>&equiv;</mo><mfenced><msub><mi>P</mi><mn>2</mn></msub></mfenced><mo>⇔</mo><mover><msub><mi>n</mi><mn>1</mn></msub><mo>→</mo></mover><mo>=</mo><mi>k</mi><mo>.</mo><mover><msub><mi>n</mi><mn>2</mn></msub><mo>→</mo></mover></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mn>1</mn></msub><mo>=</mo><mi>k</mi><mo>.</mo><msub><mi>D</mi><mn>2</mn></msub></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>P</mi><mn>1</mn></msub></mfenced></math> cắt <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>P</mi><mn>2</mn></msub></mfenced><mo>⇔</mo><mover><msub><mi>n</mi><mn>1</mn></msub><mo>→</mo></mover><mo>≠</mo><mi>k</mi><mo>.</mo><mover><msub><mi>n</mi><mn>2</mn></msub><mo>→</mo></mover></math> (nghĩa là <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>n</mi><mn>1</mn></msub><mo>→</mo></mover></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>n</mi><mn>2</mn></msub><mo>→</mo></mover></math> không cùng phương).  4. Khoảng cách từ một điểm đến một mặt phẳng. Trong không gian <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>x</mi><mi>y</mi><mi>z</mi></math> cho mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>P</mi></mfenced></math> có phương trình: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>x</mi><mo>+</mo><mi>B</mi><mi>y</mi><mo>+</mo><mi>C</mi><mi>z</mi><mo>=</mo><mn>0</mn></math> và điểm <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mn>0</mn></msub><mfenced><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>;</mo><mo> </mo><msub><mi>y</mi><mn>0</mn></msub><mo>;</mo><mo> </mo><msub><mi>z</mi><mn>0</mn></msub></mrow></mfenced></math> . Khoảng cách từ M0 đến <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>P</mi></mfenced></math>được cho bởi công thức: <span id="MathJax-Element-53-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"><mi>d</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><msub><mi>M</mi><mn>0</mn></msub></mrow><mo>,</mo><mi>P</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>A</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>x</mi><mn>0</mn></msub></mrow><mo>+</mo><mi>B</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>y</mi><mn>0</mn></msub></mrow><mo>+</mo><mi>C</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>z</mi><mn>0</mn></msub></mrow><mo>+</mo><mi>D</mi><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><msqrt><mrow class="MJX-TeXAtom-ORD"><msup><mi>A</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><msup><mi>B</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><msup><mi>C</mi><mn>2</mn></msup></mrow></msqrt></mrow></mfrac><mo>.</mo></math>"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mfenced><mrow><msub><mi>M</mi><mn>0</mn></msub><mo>,</mo><mo> </mo><mi>P</mi></mrow></mfenced><mo>=</mo><mfrac><mfenced open="|" close="|"><mrow><mi>A</mi><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mi>B</mi><msub><mi>y</mi><mn>0</mn></msub><mo>+</mo><mi>C</mi><msub><mi>z</mi><mn>0</mn></msub><mo>+</mo><mi>D</mi></mrow></mfenced><msqrt><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><msup><mi>B</mi><mn>2</mn></msup><mo>+</mo><msup><mi>C</mi><mn>2</mn></msup></msqrt></mfrac><mo>.</mo></math> 5. Góc giữa hai mặt phẳng. Cho hai mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>P</mi><mn>1</mn></msub></mfenced></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>P</mi><mn>2</mn></msub></mfenced></math> có phương trình :   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mfenced><msub><mi>P</mi><mn>1</mn></msub></mfenced><mo>:</mo><mo> </mo><msub><mi>A</mi><mn>1</mn></msub><mi>x</mi><mo>+</mo><msub><mi>B</mi><mn>1</mn></msub><mi>y</mi><mo>+</mo><msub><mi>C</mi><mn>1</mn></msub><mi>z</mi><mo>+</mo><msub><mi>D</mi><mn>1</mn></msub><mo>=</mo><mn>0</mn></math>;    <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>P</mi><mn>2</mn></msub></mfenced><mo>:</mo><mo> </mo><msub><mi>A</mi><mn>2</mn></msub><mi>x</mi><mo>+</mo><msub><mi>B</mi><mn>2</mn></msub><mi>y</mi><mo>+</mo><msub><mi>C</mi><mn>2</mn></msub><mi>z</mi><mo>+</mo><msub><mi>D</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn><mo>.</mo></math> Gọi <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi></math> là góc giữa hai mặt phẳng <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>P</mi><mn>1</mn></msub></mfenced></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>P</mi><mn>2</mn></msub></mfenced></math> thì <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>⩽</mo><mi>φ</mi><mo>⩽</mo><mn>90</mn><mo>°</mo></math><span id="MathJax-Element-60-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"><mn>0</mn><mspace width="thickmathspace" /><mo>&#x2264;</mo><mspace width="thickmathspace" /><mi>&#x03C6;</mi><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mo>&#x2264;</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><msup><mn>90</mn><mrow class="MJX-TeXAtom-ORD"><mn>0</mn><mspace width="thickmathspace" /></mrow></msup></mrow></math>"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><msup><mrow class="MJX-TeXAtom-ORD"><mspace width="thickmathspace"></mspace></mrow></msup></mrow></math> và : <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mi>φ</mi><mo>=</mo><mfenced open="|" close="|"><mrow><mi>cos</mi><mfenced><mover><mrow><mover><msub><mi>n</mi><mn>1</mn></msub><mo>→</mo></mover><mo>,</mo><mo> </mo><mover><msub><mi>n</mi><mn>2</mn></msub><mo>→</mo></mover></mrow><mo>^</mo></mover></mfenced></mrow></mfenced><mo>=</mo><mfrac><mfenced open="|" close="|"><mrow><msub><mi>A</mi><mn>1</mn></msub><msub><mi>A</mi><mn>2</mn></msub><mo>+</mo><msub><mi>B</mi><mn>1</mn></msub><msub><mi>B</mi><mn>2</mn></msub><mo>+</mo><msub><mi>C</mi><mn>1</mn></msub><msub><mi>C</mi><mn>2</mn></msub><mo>+</mo><mi>D</mi></mrow></mfenced><mrow><msqrt><msubsup><mi>A</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>B</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>C</mi><mn>1</mn><mn>2</mn></msubsup></msqrt><mo>.</mo><msqrt><msubsup><mi>A</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>B</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>C</mi><mn>2</mn><mn>2</mn></msubsup></msqrt></mrow></mfrac></math>

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