Аналитическая геометрия и линейная алгебра. Учебное пособие

ǾȇȘșȣ ǶǷǵǸǹǷǧǴǸǹǩǵ

2.1. ǸȏȘșȌȓȢ ȑȕȕȗȋȏȔȇș ȉ ȖȗȕȘșȗȇȔȘșȉȌ

Ɉɩɪɟɞɟɥɟɧɢɟ Ƚɨɜɨɪɹɬ ɱɬɨɜɬɪɟɯɦɟɪɧɨɦɩɪɨɫɬɪɚɧɫɬɜɟɡɚɞɚɧɚɞɟɤɚɪ ɬɨɜɚ ɩɪɹɦɨɭɝɨɥɶɧɚɹ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ ɟɫɥɢ

1)ɡɚɞɚɧɵɬɪɢɜɡɚɢɦɧɨɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɟɩɪɹɦɵɟ ɩɟɪɟɫɟɤɚɸɳɢɟɫɹɜɨɞ ɧɨɣ ɬɨɱɤɟ ɬɨɱɤɚ ɢɯ ɩɟɪɟɫɟɱɟɧɢɹ ɧɚɡɵɜɚɟɬɫɹ ɧɚɱɚɥɨɦ ɤɨɨɪɞɢɧɚɬ O);

2)ɧɚ ɤɚɠɞɨɣ ɢɡ ɩɪɹɦɵɯ ɜɵɛɪɚɧɨ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɧɚɩɪɚɜɥɟɧɢɟ

3)ɜɵɛɪɚɧɚ ɟɞɢɧɢɰɚ ɦɚɫɲɬɚɛɚ e.

ɍɤɚɡɚɧɧɵɟ ɜ ɨɩɪɟɞɟɥɟɧɢɢ ɩɪɹɦɵɟ ɧɚɡɵɜɚɸɬɫɹ ɤɨɨɪɞɢɧɚɬɧɵɦɢ ɨɫɹɦɢ Ɉɞɧɚɢɡ ɧɢɯɧɚɡɵɜɚɟɬɫɹɨɫɶɸɚɛɫɰɢɫɫ ɨɫɶOx ɜɬɨɪɚɹ– ɨɫɶɸɨɪɞɢɧɚɬ ɨɫɶOy ɬɪɟɬɶɹ – ɨɫɶɸ ɚɩɩɥɢɤɚɬ ɨɫɶ Oz).

ɉɭɫɬɶ M – ɬɨɱɤɚ ɩɪɨɫɬɪɚɧɫɬɜɚ Ɉɩɭɫɬɢɦ ɢɡ ɷɬɨɣ ɬɨɱɤɢ ɩɟɪɩɟɧɞɢɤɭɥɹɪɵ ɧɚ ɨɫɢ Ox Oy ɢ Oz ɉɭɫɬɶ Mx My ɢ Mz – ɬɨɱɤɢ ɩɟɪɟɫɟɱɟɧɢɹ ɩɟɪɩɟɧɞɢɤɭɥɹɪɨɜ ɫ ɤɨɨɪɞɢɧɚɬɧɵɦɢ ɨɫɹɦɢ Ⱦɟɤɚɪɬɨɜɵɦɢ ɩɪɹɦɨɭɝɨɥɶɧɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ ɬɨɱɤɢ M ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɟɤ Mx My ɢ Mz ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɨɫɹɯ ɍɩɨ ɪɹɞɨɱɟɧɧɚɹ ɬɪɨɣɤɚ (x y z) ɧɚɡɵɜɚɟɬɫɹ ɞɟɤɚɪɬɨɜɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ ɬɨɱɤɢ M.

ɉɭɫɬɶ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɚɞɚɧɨ ɞɜɟ ɬɨɱɤɢ A x1 y1]1 ɢ B x2 y2]2 Ɋɚɫɫɬɨ

Ɏɨɪɦɭɥɭ ɥɟɝɤɨ ɩɨɥɭɱɢɬɶ ɩɪɢɦɟɧɹɹ ɬɟɨɪɟɦɭ ɉɢɮɚɝɨɪɚ ɉɪɢɦɟɪɇɚɣɞɟɦ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɬɨɱɤɚɦɢ A ɢ B

AB 2 2 2

ɉɭɫɬɶ ɞɥɹ ɡɚɞɚɧɧɵɯ ɬɨɱɟɤ A x1 y1]1 ɢ B x2 y2]2 ɬɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɬɨɱɤɭ C ɥɟɠɚɳɭɸ ɧɚ ɨɬɪɟɡɤɟ >A B] ɞɟɥɹɳɭɸ ɨɬɪɟɡɨɤ ɜ ɨɬɧɨɲɟɧɢɢ M

AC

CB M

ɝɞɟ M – ɡɚɞɚɧɧɨɟ ɜɟɳɟɫɬɜɟɧɧɨɟ ɱɢɫɥɨ Ɉɛɨɡɧɚɱɢɦ ɧɟɢɡɜɟɫɬɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ C ɤɚɤ (x y z). Ɉɩɭɫɬɢɦ ɢɡ ɬɨɱɟɤ A B C ɩɟɪɩɟɧɞɢɤɭɥɹɪɵ ɧɚ ɨɫɶ Ox ɢ ɨɛɨɡɧɚɱɢɦ ɩɪɨɟɤɰɢɢ ɷɬɢɯ ɬɨɱɟɤ P1 P2 P ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɉɪɹɦɵɟ AP1 (BP2) ɢ (CP) ɩɚɪɚɥɥɟɥɶɧɵ ɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ

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AC P1P .

CB PP2

Ɉɬɪɟɡɨɤ >P1 P2@ ɩɪɢɧɚɞɥɟɠɢɬ ɤɨɨɪɞɢɧɚɬɧɨɣ ɩɪɹɦɨɣ ɢ ɤɨɨɪɞɢɧɚɬɭ x ɬɨɱ ɤɢ P ɞɟɥɹɳɟɣ ɟɝɨ ɜ ɨɬɧɨɲɟɧɢɢ M ɦɨɠɧɨ ɧɚɣɬɢ ɩɨ ɮɨɪɦɭɥɟ

x x1 Mx2 . 1 M

ɇɚɣɞɟɧɧɚɹ ɜɟɥɢɱɢɧɚ ɹɜɥɹɟɬɫɹ ɢɫɤɨɦɨɣ ɚɛɫɰɢɫɫɨɣ ɬɨɱɤɢ C Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɧɨ ɧɚɣɬɢ ɜɬɨɪɭɸ ɢ ɬɪɟɬɶɸ ɤɨɨɪɞɢɧɚɬɵ ɷɬɨɣ ɬɨɱɤɢ Ɍ ɨ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ C ɧɚɯɨɞɢɦ ɩɨ ɮɨɪɦɭɥɚɦ

ɉɪɢɦɟɪɉɭɫɬɶ A B ɇɚɣɞɟɦ ɫɟɪɟɞɢɧɭ ɨɬɪɟɡɤɚ >A B]. ɉɨɞɫɬɚɜɥɹɹ ɜ ɮɨɪɦɭɥɭ M 1 ɢ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɟɤ A ɢ B ɩɨɥɭɱɚɟɦ ɤɨɨɪ

Ɋɚɫɫɦɨɬɪɢɦ ɞɪɭɝɨɣ ɫɩɨɫɨɛ ɩɨɫɬɪɨɟɧɢɹ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɜ ɬɪɟɯɦɟɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ

ɉɭɫɬɶ ɜ ɬɪɟɯɦɟɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɜɵɛɪɚɧɚ ɩɥɨɫɤɨɫɬɶ B ɢ ɩɟɪɩɟɧɞɢɤɭ ɥɹɪɧɚɹ ɤ ɷɬɨɣ ɩɥɨɫɤɨɫɬɢ ɨɫɶ Oz ȼɵɛɟɪɟɦ ɜ ɩɥɨɫɤɨɫɬɢ B ɥɭɱ Ox ɫ ɧɚɱɚɥɨɦ ɜ ɬɨɱɤɟ O ɩɟɪɟɫɟɱɟɧɢɹ ɨɫɢ Oz ɫ ɩɥɨɫɤɨɫɬɶɸ ɐɢɥɢɧɞɪɢɱɟɫɤɢɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ ɩɪɨɢɡɜɨɥɶɧɨɣ ɬɨɱɤɢ P ɩɪɨɫɬɪɚɧɫɬɜɚ ɧɚɡɵɜɚɸɬɫɹ ɱɢɫɥɚ S K ɢ z ɩɟɪɜɵɟ ɞɜɚ ɢɡ ɤɨɬɨɪɵɯ ɹɜɥɹɸɬɫɹ ɩɨɥɹɪɧɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ ɩɪɨɟɤɰɢɢ ɬɨɱɤɢ M ɜ ɩɥɨɫ ɤɨɫɬɶ B ɱɢɫɥɨ z – ɤɨɨɪɞɢɧɚɬɚ ɩɪɨɟɤɰɢɢ M ɧɚ ɨɫɶ Oz.

ǷȏȘ

ȼɵɛɟɪɟɦ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɞɟɤɚɪɬɨɜɭ ɩɪɹɦɨɭɝɨɥɶɧɭɸ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɫɜɹɡɚɧɧɭɸ ɫ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɨɣ ɤɨɨɪɞɢɧɚɬ ɨɫɢ Oz

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ɜ ɨɛɟɢɯɫɢɫɬɟɦɚɯɫɨɜɩɚɞɚɸɬ ɩɨɥɹɪɧɵɣɥɭɱOx ɜɩɥɨɫɤɨɫɬɢ B ɥɟɠɢɬ ɧɚɨɫɢ Ox ɨɫɶ Oy ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɬɨɱɤɭ O ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɤ ɨɫɢ Ox Ɍɨɝɞɚ ɞɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ P ɫɜɹɡɚɧɵ ɫ ɟɟ ɰɢɥɢɧɞɪɢɱɟɫɤɢɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ ɫɨɨɬɧɨɲɟ ɧɢɹɦɢ

ɇɚɣɞɟɦ ɰɢɥɢɧɞɪɢɱɟɫɤɢɟ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ ɡɚɞɚɧɧɨɣ ɫɜɨɢɦɢ ɞɟɤɚɪɬɨ ɜɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ M ɉɨɞɫɬɚɜɥɹɟɦ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ ɜ ɮɨɪɦɭ ɥɵ

ȿɳɟɨɞɧɢɦ ɫɩɨɫɨɛɨɦ ɡɚɞɚɬɶɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ ɬɪɟɯɦɟɪɧɨɝɨɩɪɨɫɬɪɚɧɫɬɜɚ ɹɜɥɹɟɬɫɹ ɫɮɟɪɢɱɟɫɤɚɹ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ.

ɉɭɫɬɶ ɜɬɪɟɯɦɟɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɡɚɞɚɧɵ ɬɪɢɜɡɚɢɦɧɨɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɟ ɨɫɢ Ox Oy ɢ Oz ɫ ɨɛɳɢɦ ɧɚɱɚɥɨɦ ɜ ɬɨɱɤɟ O ɉɭɫɬɶ P – ɩɪɨɢɡɜɨɥɶɧɚɹ ɬɨɱɤɚ ɩɪɨɫɬɪɚɧɫɬɜɚ ɢ N – ɟɟ ɩɪɨɟɤɰɢɹ ɜ ɩɥɨɫɤɨɫɬɶ Oxy.

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ɇɚɣɞɟɦ ɬɨɱɤɭ A1 r1 K1 R1 ɫɢɦɦɟɬɪɢɱɧɭɸ ɬɨɱɤɟ A r K R ɨɬɧɨɫɢɬɟɥɶɧɨ

ɉɭɫɬɶɬɨɱɤɚ A6 r6 K6 R6 ɫɢɦɦɟɬɪɢɱɧɚɬɨɱɤɟ A r K R ɨɬɧɨɫɢɬɟɥɶɧɨɩɥɨɫ ɤɨɫɬɢ Oxz ɋɮɟɪɢɱɟɫɤɢɣ ɪɚɞɢɭɫ r6 r 3. K6 K Q4 . ɍɝɨɥ R6 R Q6 .

Ɍ ɨ ɫɮɟɪɢɱɟɫɤɢɟ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ A6

Ɍɟɤɫɬ ɡɚɞɚɧɢɹ ʋ Ɍɨɱɤɚ A r K R ɡɚɞɚɧɚ ɫɜɨɢɦɢ ɫɮɟɪɢɱɟɫɤɢɦɢ ɤɨ ɨɪɞɢɧɚɬɚɦɢ ɇɚɣɬɢ ɞɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚɬɵ ɷɬɨɣ ɬɨɱɤɢ ɇɚɣɬɢ ɫɮɟɪɢɱɟɫɤɢɟ ɤɨ ɨɪɞɢɧɚɬɵ ɬɨɱɟɤ ɫɢɦɦɟɬɪɢɱɧɵɯ ɡɚɞɚɧɧɨɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɚ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ ɛ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ ɜ ɤɨɨɪɞɢɧɚɬɧɵɯ ɩɥɨɫɤɨɫɬɟɣ

ȼɚɪɢɚɧɬɵ ɡɚɞɚɧɢɹ ʋ

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2.2. ǩȌȑșȕȗȢ

ɇɟɫɦɨɬɪɹ ɧɚ ɬɨ ɱɬɨ ɱɢɬɚɬɟɥɶ ɯɨɪɨɲɨ ɡɧɚɤɨɦ ɫ ɩɨɧɹɬɢɟɦ ɜɟɤɬɨɪɚ ɢɡ ɲɤɨɥɶɧɨɝɨ ɤɭɪɫɚ ɦɚɬɟɦɚɬɢɤɢ ɧɚɩɨɦɧɢɦ ɨɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ ɫɜɹɡɚɧɧɵɟ ɫ ɞɚɧ ɧɵɦ ɝɟɨɦɟɬɪɢɱɟɫɤɢɦ ɨɛɴɟɤɬɨɦ

Ɉɩɪɟɞɟɥɟɧɢɟ Ƚɟɨɦɟɬɪɢɱɟɫɤɢɦɜɟɤɬɨɪɨɦɛɭɞɟɦɧɚɡɵɜɚɬɶɧɚɩɪɚɜɥɟɧ

ɜɟɤɬɨɪɚ ȼɟɤɬɨɪ ɧɚɡɵɜɚɸɬ ɧɭɥɟɜɵɦ ɟɫɥɢ ɟɝɨ ɧɚɱɚɥɨ ɢ ɤɨɧɟɰ ɫɨɜɩɚɞɚɸɬ ɇɭɥɟɜɨɣ

ɜɟɤɬɨɪ ɧɟ ɢɦɟɟɬ ɧɚɩɪɚɜɥɟɧɢɹ ɢ ɢɦɟɟɬ ɞɥɢɧɭ ɪɚɜɧɭɸ ɧɭɥɸ ȼɟɤɬɨɪɵ ɧɚɡɵɜɚɸɬɫɹ ɤɨɥɥɢɧɟɚɪɧɵɦɢ ɟɫɥɢ ɨɧɢ ɥɟɠɚɬ ɧɚ ɩɚɪɚɥɥɟɥɶɧɵɯ

ɩɪɹɦɵɯ Ⱦɜɚ ɜɟɤɬɨɪɚ ɧɚɡɵɜɚɸɬɫɹ ɪɚɜɧɵɦɢ ɟɫɥɢ ɨɧɢ ɤɨɥɥɢɧɟɚɪɧɵɟ ɢɦɟɸɬ

4)ɞɥɹ ɥɸɛɨɝɨ ɜɟɤɬɨɪɚ a ɫɭɳɟɫɬɜɭɟɬ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɣ ɜɟɤɬɨɪ aa ɬɚ ɤɨɣ ɱɬɨ

G JG G a aa 0.

2) (B C)a Ba Ca ;

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