Hướng dẫn giải Bài 3 (Trang 68 SGK Toán Hình học 12)

Cho hình hộp ABCD.A'B'C'D' biết <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfenced><mrow><mn>1</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>1</mn></mrow></mfenced><mo>,</mo><mo> </mo><mi>B</mi><mo>=</mo><mfenced><mrow><mn>2</mn><mo>;</mo><mn>1</mn><mo>;</mo><mn>2</mn></mrow></mfenced><mo>,</mo><mo> </mo><mi>D</mi><mo>=</mo><mfenced><mrow><mn>1</mn><mo>;</mo><mo>-</mo><mn>1</mn><mo>;</mo><mn>1</mn></mrow></mfenced><mo>,</mo><mo> </mo><mi>C</mi><mo>'</mo><mo>=</mo><mfenced><mrow><mn>4</mn><mo>;</mo><mn>5</mn><mo>;</mo><mo>-</mo><mn>5</mn></mrow></mfenced></math>. Tính tọa độ các đỉnh còn lại của hình hộp. Giải <img class="wscnph" src="https://static.colearn.vn:8413/v1.0/upload/library/18022022/screenshot-191-FlLOK3.png" width="405" height="288" /> 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><mn>1</mn><mo>;</mo><mn>1</mn><mo>;</mo><mn>1</mn></mrow></mfenced><mspace linebreak="newline"/><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mn>0</mn><mo>;</mo><mo>-</mo><mn>1</mn><mo>;</mo><mn>0</mn></mrow></mfenced><mspace linebreak="newline"/><mover><mrow><mi>B</mi><mi>C</mi></mrow><mo>→</mo></mover><mo>=</mo><mover><mrow><mi>A</mi><mi>D</mi></mrow><mo>→</mo></mover><mo>⇔</mo><mfenced open="{" close=""><mrow><mtable columnalign="left"><mtr><mtd><msub><mi>x</mi><mi>C</mi></msub><mo>-</mo><mn>2</mn><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><msub><mi>y</mi><mi>C</mi></msub><mo>-</mo><mn>1</mn><mo>=</mo><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><msub><mi>z</mi><mi>C</mi></msub><mo>-</mo><mn>2</mn><mo>=</mo><mn>0</mn></mtd></mtr></mtable><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><msub><mi>x</mi><mi>C</mi></msub><mo>=</mo><mn>2</mn></mtd></mtr><mtr><mtd><msub><mi>y</mi><mi>C</mi></msub><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd><msub><mi>z</mi><mi>C</mi></msub><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math> Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mo>(</mo><mn>2</mn><mo>;</mo><mn>0</mn><mo>;</mo><mn>2</mn><mo>)</mo></math> Suy ra <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>C</mi><mi>C</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>=</mo><mo>(</mo><mn>2</mn><mo>;</mo><mn>5</mn><mo>;</mo><mo>-</mo><mn>7</mn><mo>)</mo></math> Từ <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>A</mi><mi>A</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>=</mo><mover><mrow><mi>B</mi><mi>B</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>=</mo><mover><mrow><mi>D</mi><mi>D</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>=</mo><mover><mrow><mi>C</mi><mi>C</mi><mo>'</mo></mrow><mo>→</mo></mover><mo>=</mo><mfenced><mrow><mn>2</mn><mo>;</mo><mn>5</mn><mo>;</mo><mo>-</mo><mn>7</mn></mrow></mfenced></math> Suy ra <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><msub><mi>x</mi><mrow><mi>A</mi><mo>'</mo></mrow></msub><mo>-</mo><mn>1</mn><mo>=</mo><mn>2</mn></mtd></mtr><mtr><mtd><msub><mi>y</mi><mrow><mi>A</mi><mo>'</mo></mrow></msub><mo>-</mo><mn>0</mn><mo>=</mo><mn>5</mn></mtd></mtr><mtr><mtd><msub><mi>z</mi><mrow><mi>A</mi><mo>'</mo></mrow></msub><mo>-</mo><mn>1</mn><mo>=</mo><mo>-</mo><mn>7</mn></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><msub><mi>x</mi><mrow><mi>A</mi><mo>'</mo></mrow></msub><mo>=</mo><mn>3</mn></mtd></mtr><mtr><mtd><msub><mi>y</mi><mrow><mi>A</mi><mo>'</mo></mrow></msub><mo>=</mo><mn>5</mn></mtd></mtr><mtr><mtd><msub><mi>z</mi><mrow><mi>A</mi><mo>'</mo></mrow></msub><mo>=</mo><mo>-</mo><mn>6</mn></mtd></mtr></mtable></mfenced></math> Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>'</mo><mo>=</mo><mo>(</mo><mn>3</mn><mo>;</mo><mn>5</mn><mo>;</mo><mo>-</mo><mn>6</mn><mo>)</mo></math>. Tương tự <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>'</mo><mo>=</mo><mo>(</mo><mn>4</mn><mo>;</mo><mn>6</mn><mo>;</mo><mo>-</mo><mn>5</mn><mo>)</mo><mo>,</mo><mo> </mo><mi>D</mi><mo>'</mo><mo>=</mo><mo>(</mo><mn>3</mn><mo>;</mo><mn>4</mn><mo>;</mo><mo>-</mo><mn>6</mn><mo>)</mo></math>