Ploshina odne z osnovnih ponyat geometriyi Pri sistematichnomu vikladenni geometriyi ponyattya ploshini yak pravilo sprijmayetsya yak pervisne kotre lishe oposeredkovano viznachayetsya aksiomami geometriyi Rivnyannya ploshini vpershe zustrichayetsya v A K Klero 1731 rivnyannya ploshini u vidrizkah vochevid vpershe zustrichayetsya v Lame 1816 1818 normalne rivnyannya uviv 1861 Dvi ploshini sho peretinayutsya Zmist 1 Deyaki harakterni vlastivosti ploshini 2 Ploshini v trivimirnomu Evklidovomu prostori 2 1 Viznachennya na osnovi tochok i pryamih sho nalezhat ploshini 2 2 Vlastivosti 3 Rivnyannya ploshini 4 Pov yazani ponyattya 5 Literatura 6 PosilannyaDeyaki harakterni vlastivosti ploshini RedaguvatiPloshina poverhnya kotra povnistyu mistit kozhnu pryamu sho spoluchaye yiyi dovilni tochki Ploshina mnozhina tochok rivnoviddalenih vid dvoh zadanih Ploshini v trivimirnomu Evklidovomu prostori RedaguvatiViznachennya na osnovi tochok i pryamih sho nalezhat ploshini Redaguvati V Evklidovomu prostori bud yakoyi vimirnosti ploshina zazvichaj viznachayetsya za dopomogoyu Troh ne kolinearnih tochok tochki ne znahodyatsya na odnij pryamij Pryamoyu i tochkoyu sho ne nalezhit cij pryamij Dvoma riznimi pryamimi sho peretinayutsya Dvoma paralelnimi pryamimi Vlastivosti Redaguvati Nastupni tverdzhennya spravedlivi dlya trivimirnogo Evklidovogo prostoru ale ne dlya bilshih rozmirnostej hocha voni mayut analogiyi pri vishih rozmirnostyah Dvi rizni ploshini ye abo paralelnimi abo peretinayutsya po pryamij Pryama mozhe buti abo paralelnoyu do ploshini abo peretinaye yiyi v yedinij tochci abo znahoditsya na ploshini Dvi rizni pryami perpendikulyarni do odniyeyi ploshini mayut buti paralelnimi odna do odnoyi Dvi rizni ploshini perpendikulyarni odnij pryamij mayut buti paralelni odna odnij Rivnyannya ploshini RedaguvatiPloshina algebrichna poverhnya pershogo poryadku v dekartovij sistemi koordinat ploshina mozhe buti zadana rivnyannyam pershogo stepenya Zagalne povne rivnyannya ploshiniA x B y C z D 0 1 displaystyle Ax By Cz D 0 qquad 1 nbsp de A B C displaystyle A B C nbsp ta D displaystyle D nbsp stali pri chomu A B displaystyle A B nbsp i C displaystyle C nbsp ne vsi rivni nulyu u vektornij formi r N D 0 displaystyle mathbf r mathbf N D 0 nbsp de r displaystyle mathbf r nbsp radius vektor tochki M x y z displaystyle M x y z nbsp vektor N A B C displaystyle mathbf N A B C nbsp perpendikulyarnij do ploshini normalnij vektor Napryamni kosinusi vektora N displaystyle mathbf N nbsp cos a A A 2 B 2 C 2 displaystyle cos alpha frac A sqrt A 2 B 2 C 2 nbsp cos b B A 2 B 2 C 2 displaystyle cos beta frac B sqrt A 2 B 2 C 2 nbsp cos g C A 2 B 2 C 2 displaystyle cos gamma frac C sqrt A 2 B 2 C 2 nbsp Yaksho odin z koeficiyentiv v rivnyanni ploshini dorivnyuye nulyu rivnyannya nazivayetsya nepovnim Pri D 0 displaystyle D 0 nbsp ploshina prohodit cherez pochatok koordinat pri A 0 displaystyle A 0 nbsp abo B 0 displaystyle B 0 nbsp C 0 displaystyle C 0 nbsp ploshina paralelna osi O x displaystyle Ox nbsp vidpovidno O y displaystyle Oy nbsp chi O z displaystyle Oz nbsp Pri A B 0 displaystyle A B 0 nbsp A C 0 displaystyle A C 0 nbsp chi B C 0 displaystyle B C 0 nbsp ploshina paralelna ploshini O x y displaystyle Oxy nbsp vidpovidno O x z displaystyle Oxz nbsp chi O y z displaystyle Oyz nbsp Rivnyannya ploshini u vidrizkah x a y b z c 1 displaystyle frac x a frac y b frac z c 1 nbsp de a D A b D B c D C displaystyle a D A b D B c D C nbsp vidrizki yaki ploshina vidsikaye na osyah O x O y displaystyle Ox Oy nbsp i O z displaystyle Oz nbsp Rivnyannya ploshini sho prohodit cherez tochku M x 0 y 0 z 0 displaystyle M x 0 y 0 z 0 nbsp perpendikulyarno do vektora N A B C displaystyle mathbf N A B C nbsp A x x 0 B y y 0 C z z 0 0 displaystyle A x x 0 B y y 0 C z z 0 0 nbsp u vektornij formi r r 0 N 0 displaystyle mathbf r mathbf r 0 mathbf N 0 nbsp Rivnyannya ploshini sho prohodit cherez tri zadani tochki M x i y i z i displaystyle M x i y i z i nbsp yaki ne lezhat na odnij pryamij r r 1 r r 2 r r 3 0 displaystyle mathbf r mathbf r 1 mathbf r mathbf r 2 mathbf r mathbf r 3 0 nbsp mishanij dobutok vektoriv inshimi slovami x x 1 y y 1 z z 1 x 2 x 1 y 2 y 1 z 2 z 1 x 3 x 1 y 3 y 1 z 3 z 1 0 displaystyle left begin matrix x x 1 amp y y 1 amp z z 1 x 2 x 1 amp y 2 y 1 amp z 2 z 1 x 3 x 1 amp y 3 y 1 amp z 3 z 1 end matrix right 0 nbsp Normalne normovane rivnyannya ploshinix cos a y cos b z cos g p 0 2 displaystyle x cos alpha y cos beta z cos gamma p 0 qquad 2 nbsp u vektornij formi r N 0 0 displaystyle mathbf r mathbf N 0 0 nbsp de N 0 displaystyle mathbf N 0 nbsp odinichnij vektor p displaystyle p nbsp vidstan vid ploshini do pochatku koordinat Rivnyannya 2 mozhna otrimati z rivnyannya 1 pomnozhivshi jogo na normuyuchij mnozhnik m 1 A 2 B 2 C 2 displaystyle mu pm frac 1 sqrt A 2 B 2 C 2 nbsp znaki m displaystyle mu nbsp i D displaystyle D nbsp protilezhni Pov yazani ponyattya RedaguvatiVidhilennya tochki M 1 x 1 y 1 z 1 displaystyle M 1 x 1 y 1 z 1 nbsp vid ploshinid x 1 cos a y 1 cos b z 1 cos g p displaystyle delta x 1 cos alpha y 1 cos beta z 1 cos gamma p nbsp d gt 0 displaystyle delta gt 0 nbsp yaksho M i displaystyle M i nbsp i pochatok koordinat lezhat po rizni storoni ploshini v protilezhnomu vipadkud lt 0 displaystyle delta lt 0 nbsp Vidstan vid tochki do ploshini dorivnyuye d displaystyle delta nbsp Kut mizh ploshinami Yaksho rivnyannya ploshini zadani u viglyadi 1 tocos f A 1 A 2 B 1 B 2 C 1 C 2 A 1 2 B 1 2 C 1 2 A 2 2 B 2 2 C 2 2 displaystyle cos varphi frac A 1 A 2 B 1 B 2 C 1 C 2 sqrt A 1 2 B 1 2 C 1 2 A 2 2 B 2 2 C 2 2 nbsp Yaksho u vektornij formi to cos f N 1 N 2 N 1 N 2 displaystyle cos varphi frac mathbf N 1 mathbf N 2 mathbf N 1 mathbf N 2 nbsp Ploshini paralelni yakshoA 1 A 2 B 1 B 2 C 1 C 2 displaystyle frac A 1 A 2 frac B 1 B 2 frac C 1 C 2 nbsp chi N 1 N 2 1 displaystyle mathbf N 1 mathbf N 2 1 nbsp Ploshini perpendikulyarni yakshoA 1 A 2 B 1 B 2 C 1 C 2 0 displaystyle A 1 A 2 B 1 B 2 C 1 C 2 0 nbsp chi N 1 N 2 0 displaystyle mathbf N 1 mathbf N 2 0 nbsp Puchok ploshin rivnyannya dovilnoyi ploshini sho prohodit cherez liniyu peretinu dvoh ploshina A 1 x B 1 y C 1 z b A 2 x B 2 y C 2 z 0 displaystyle alpha A 1 x B 1 y C 1 z beta A 2 x B 2 y C 2 z 0 nbsp de a displaystyle alpha nbsp i b displaystyle beta nbsp dovilni chisla sho ne odnochasno dorivnyuyut nulyu Literatura RedaguvatiIlin V A Poznyak E G Analiticheskaya geometriya M FIZMATLIT 2002 r 240s Posilannya RedaguvatiPloshina Visha matematika v prikladah i zadachah Klepko V Yu Golec V L 2 ge vidannya K Centr uchbovoyi literaturi 2009 S 131 594 s nbsp Portal Matematika Otrimano z https uk wikipedia org w index php title Ploshina amp oldid 37008378
