Mukana wokuti vanhu vanga ndege: sei kuita? Types Ndege equations

The ndege nzvimbo inogona kurondedzerwa nenzira dzakasiyana (rimwe dot uye Vector, kuti vektori maviri pfungwa, matatu pfungwa, etc.). Ndiwo mupfungwa, ndege mukana wokuti vanhu vanga anogona kuva mhando dzakasiyana. Uyewo pasi zvimwe ezvinhu ndege kufananidza, perpendicular, intersecting, etc. On ichi uye vanotaura munyaya ino. Tichadzidza kuti mukuru mukana wokuti vanhu vanga ndege uye kwete chete.

The zvinoitika chimiro mukana wokuti vanhu vanga

Ngatitii R ndiye nzvimbo _3, iro rine rectangular batanidza hurongwa xyz. Tinotsanangura Vector α, izvo vachasunungurwa pokutangira O. Kuburikidza kuguma Vector α kunochera ndege P riri perpendicular nayo.

Kureva P panguva imwe zvemasanga nguva Q = (x, y, z). The nharaunda Vector yepfungwa Q chiratidzo tsamba p. Kureba Vector vakaenzana α p = IαI uye Ʋ = (cosα, cosβ, cosγ).

Ichi chikwata Vector, iyo chinotungamirwa nenzira sezvo vector α. α, β uye γ - ndivo pamakona kuti akaumba pakati Vector uye zvakanaka mirayiridzo Ʋ nzvimbo matemo X, y, z azvaitwa. The fungidziro ane pfungwa iri Vector QεP Ʋ ari kugara izvo zvakaenzana S. (p, Ʋ) = peji (r≥0).

The pamusoro mukana wokuti vanhu vanga kune zvakunoreva apo p = 0. The chete N ndege iyi, kuti vayambuke pfungwa O (α = 0), rinova mavambo, uye chikwata Vector Ʋ, asunungurwa kubva pfungwa O achava perpendicular kuna P, kunyange zvayo mirayiridzo, izvo zvinoreva kuti vector Ʋ tsunga kusvikira chiratidzo. Previous mukana wokuti vanhu vanga ndiye ndege yedu P, runoratidzwa vector fomu. Asi tichifunga Coordonatele ndiko:

P mukuru kana kuenzana 0. Tawana ndege mukana wokuti vanhu vanga chaiye chimiro.

The vakawanda mukana wokuti vanhu vanga

Kana mukana wokuti vanhu vanga vari Coordonatele zviwande kubudikidza chero nhamba kuti haana kuenzana zero, isu tiwane mukana wokuti vanhu vanga yakaenzana ichi kuti inorondedzera chaizvo ndege. It achava zvinotevera chimiro:

Pano, A, B, C - inhamba panguva dzakasiyana razero. mukana wokuti vanhu vanga ichi anonzi mukana wokuti vanhu vanga ari mukuru chimiro ndege.

The equations of ndege. Special zviitiko

Mukana wokuti vanhu vanga anogona kazhinji dzakachinjwa Zvimwe ezvinhu. Funga zvimwe vavo.

Kufunga kuti coefficient A iri 0. Izvi zvinoratidza kuti ndege akaererana zvakafanorongwa akabatana Ox. Panyaya iyi, chimiro mukana wokuti vanhu vanga anochinja: Wu + CI + D = 0.

Saizvozvowo, chimiro mukana wokuti vanhu vanga uye achizosiyana pamwe anotevera ezvinhu:

• Chekutanga, kana B = 0, mukana wokuti vanhu vanga kuchinja kuti Ax + CI + D = 0, izvo zvaizoita kuratidza parallelism kuti akabatana Oy.
• Chechipiri, kana C = 0, mukana wokuti vanhu vanga iri kuva Ax + By + D = 0, kureva pamusoro akaererana zvakafanorongwa akabatana Oz.
• Chechitatu, kana D = 0, mukana wokuti vanhu vanga achaonekwa sezvo Ax + By + CI = 0, izvo zvinoreva kuti ndege intersects O (mavambo).
• Chechina, kana A = B = 0, mukana wokuti vanhu vanga kuchinja kuti CI + D = 0, izvo zvicharatidza parallelism Oxy.
• Chechishanu, kana B = C = 0, mukana wokuti vanhu vanga inova Ax + D = 0, izvo zvinoreva kuti ndege zvinofambirana Oyz.
• Sixthly, kana A = C = 0, mukana wokuti vanhu vanga anotora chimiro Wu + D = 0, i.e., zvichaita vaudze parallelism Oxz.

Chimiro mukana wokuti vanhu vanga vari zvikamu

Kana apo nhamba A, B, C, D akasiyana zero, muchimiro mukana wokuti vanhu vanga (0) angava sezvinotevera:

X / a + Y / b + z / C = 1,

makavigwa munhu = -D / A, b = -D / B, c = -D / C.

We semubairo mukana wokuti vanhu vanga ndege zvidimbu. Zvinofanira kucherechedzwa kuti ndege ino piringisha kuti X-akabatana panguva pamwe Coordonatele (a, 0,0), Oy - (0, b, 0), uye Oz - (0,0, s).

Akapiwa mukana wokuti vanhu vanga X / a + Y / b + z / C = 1, hazvina kuoma kuona placement Ndege hama kuti akafanorongerwa batanidza hurongwa.

The Coordinates of kwagara vector

Inowanzotorwa Vector N kuti ndege P ane Coordonatele kuti vari coefficients ari mukuru mukana wokuti vanhu vanga vari mundege, i.e. N (A, B, C).

Kuti kuona Coordonatele ari kwagara N, zvakakwana kuziva vakawanda mukana wokuti vanhu vanga kupiwa ndege.

Kana kushandisa mukana wokuti vanhu vanga vari zvikamu, chine chimiro X / a + Y / b + z / C = 1, sezvo apo uchishandisa vakawanda mukana wokuti vanhu vanga anogona kunyorwa Coordonatele chero chakaipa Vector rakapihwa ndege: (1 / a + 1 / b + 1 / c).

Zvinofanira kucherechedzwa kuti chakaipa Vector rokubatsira kugadzirisa matambudziko akasiyana-siyana. The matambudziko Zvinonyanyozivikanwa wakaumbwa uchapupu perpendicular kana dzakafanana ndege, basa kuwana pamakona pakati ndege kana pamakona pakati ndege uye yakarurama nemitsetse.

Nyorai maererano ndege mukana wokuti vanhu vanga uye Coordonatele yepfungwa chakaipa vector

A nonzero Vector N, perpendicular kuti kupiwa ndege, hunonzi (chakaipa) kune akafanorongerwa ndege.

Tomboti batanidza nzvimbo (a rectangular batanidza maitiro) Oxyz kuisa:

• Mₒ pfungwa Coordonatele (hₒ, uₒ, zₒ);
• razero Vector n = A * Ini + B * J + C * k.

Unofanira kuita mukana wokuti vanhu vanga ndege kuti ichipfuura Mₒ pfungwa perpendicular kune chakaipa n.

Mune nzvimbo yatinosarudza chero zvemasanga pfungwa uye kureva M (x, y, z). Regai nharaunda Vector neimwe pfungwa M (x, y, z) achava R = X * Ndichaisa + and * J + z * K, uye nharaunda Vector ane pfungwa Mₒ (hₒ, uₒ, zₒ) - rₒ = hₒ * Ini + uₒ * J + zₒ * k. Pfungwa M ndezvaAroni pamamiriro ndege, kana Vector MₒM kuva perpendicular kuti vector n. Tinonyora mamiriro orthogonality kushandisa scalar chigadzirwa:

[MₒM, N] = 0.

Sezvo MₒM = R-rₒ, ari Vector mukana wokuti vanhu vanga ndege vachatarira seizvi:

[R - rₒ, N] = 0.

mukana wokuti vanhu vanga Izvi zvinogona kuvawo mumwe chimiro. Nokuti chinangwa ichi, kuti zvivako zviri scalar chigadzirwa, uye vatendeuke rworuboshwe divi mukana wokuti vanhu vanga. [R - rₒ, N] = [R, N] - [rₒ, N]. Kana [rₒ, N] runotaurwa sezvo s, tinowana zvinotevera mukana wokuti vanhu vanga: [R, N] - a = 0 kana [R, N] = s, iyo inotaura nguva dzose pakati pamaberere pamusoro dzose Vector pamusoro nharaunda-vectors pamusoro apa pfungwa kuti ndezvaJehovha ndege.

Zvino unogona kuwana rinoronga mhando kurekodha ndege yedu Vector mukana wokuti vanhu vanga [R - rₒ, N] = 0. Sezvo R-rₒ = (x-hₒ) * Ini + (Y-uₒ) * J + (Z-zₒ) * K, uye n = A * ini + B * J + C * K, tine:

Zvakaratidza kuti tine mukana wokuti vanhu vanga aumbwa Ndege nomunyika pfungwa perpendicular kune dzose N:

A * (x hₒ) + B * (Y uₒ) S * (Z-zₒ) = 0.

Nyorai maererano ndege mukana wokuti vanhu vanga uye Coordonatele zviviri pfungwa yacho vector ndege collinear

Tinotsanangura mbiri zvemasanga pfungwa M '(x', y ', z') uye M "(x", y ", z"), uyewo Vector (a ', a ", mumwe ‴).

Zvino tinogona kunyora mukana wokuti vanhu vanga zvakafanorongwa ndege iyo ichipfuura aivapo pfungwa M 'uye M ", uye pfungwa imwe neimwe pamwe Coordonatele M (x, y, z) nerokuti kuti kupiwa vector.

Nokudaro M'M vectors x = {x ', y-Y'; zz '} uye M "M = {X" -x', y ', y'; z "-z '} anofanira coplanar pamwechete vector mumwe = (a ', a ", mumwe ‴), izvo zvinoreva kuti (M'M M" M, a) = 0.

Saka mukana wokuti vanhu vanga yedu ndege muchadenga vachatarira seizvi:

Mhando Ndege mukana wokuti vanhu vanga, achiyambuka nhatu pfungwa

Toti tine nhatu dzinotevera: (x ', y', z '), (x', y ', z'), (x ‴ Have ‴, z ‴), izvo hamuzi chete mutsetse. Zvakakosha kunyora mukana wokuti vanhu vanga ndege achipfuura vatatu pfungwa kutaurwa. geometry dzidziso anoti kuti ndege mhando iyi ruriko, zviri chete uye chete. Sezvo ndege iyi intersects pfungwa (x ', y', z '), ayo mukana wokuti vanhu vanga chimiro yaizova:

Pano, A, B, C zvakasiyana razero panguva imwe chete. Uyewo Ndege kupiwa intersects zvakawanda pfungwa mbiri (x ", y", z ") uye (x ‴, y ‴, z ‴). Panyaya iyi yaifanira kubva ezvinhu mhando iyi:

Zvino tinogona kusika yunifomu gadziriro pamusoro equations (Linear) pamwe unknowns nepfambi, V, w:

In yedu nyaya X, y kana z inomira zvemasanga pfungwa iyo zvinogutsa mukana wokuti vanhu vanga (1). Kufunga mukana wokuti vanhu vanga (1) uye kwegadziriro equations (2) uye (3) mamiriro ezvinhu equations zvinoratidzwa nhamba kumusoro, Vector zvinogutsa N (A, B, C) inova nontrivial. Nekuti determinant wegadziriro ndiyo razero.

Mukana wokuti vanhu vanga (1) kuti tine, ichi ndicho mukana wokuti vanhu vanga ndege. 3 nguva zvechokwadi anoenda, uye zviri nyore kuti tione. Kuti aite izvi, isu ikure determinant pedyo okunze ari nekabhunakeri. Of aripo ehupfumi determinant anotevera kuti ndege yedu panguva intersects vatatu pakutanga zvakafanorongwa pfungwa (x ', y', z '), (x ", y", z "), (x ‴, y ‴, z ‴). Saka takasarudza kuti basa pamberi pedu.

Dihedral angled pakati ndege

Dihedral angled chinhu spatial geometric chimiro inoumbwa maviri hafu-ndege kunobva achirurama mutsetse. Nemamwe mashoko, chikamu nzvimbo iyo kungogumira hafu-ndege.

Ngatitii tine ndege mbiri pamwe anotevera equations:

Tinoziva kuti Vector N = (A, B, C) uye N¹ = (A¹, H¹, S¹) maererano zvakafanorongwa ndege vari perpendicular. Panyaya iyi, pakona φ pakati vectors N uye N¹ wakaenzana angled (dihedral), iro riri pakati ndege izvi. The scalar chigadzirwa kunopiwa:

NN¹ = | N || N¹ | Kosi φ,

chaizvo nokuti

cosφ = NN¹ / | N || N¹ | = (AA¹ + VV¹ SS¹ +) / ((√ (A² + s² + V²)) * (√ (A¹) ² + (H¹) ² + (S¹) ²)).

Zvaringana kufunga kuti 0≤φ≤π.

Chaizvoizvo zviviri ndege kuti piringisha, chimiro maviri akachinjikana (dihedral): φ ₁ uye φ _2. Verengai wavo akaenzana π (φ ₁ + φ ₂ = π). Kana cosines dzavo, mhedziso tsika dzavo vakaenzana, asi ivo vari zviratidzo zvakasiyana, kureva Kosi φ ₁ = -cos φ _2. Kana mukana wokuti vanhu vanga (0) kuchatsiviwa A, B uye C -Chinhu, -B uye -C azvaitwa, mukana wokuti vanhu vanga, tinowana, achasarudza chete ndege, chete angled φ mune mukana wokuti vanhu vanga Kosi φ = NN ¹ / | N || N ¹ | It kuchatsiviwa π-φ.

Mukana wokuti vanhu vanga of perpendicular ndege

Akadana perpendicular ndege, pakati iyo pakona iri madhigirii 90. Uchishandisa akurukurwa kumusoro, tinogona kuwana mukana wokuti vanhu vanga ane ndege perpendicular kune vamwe. Ngatitii tine ndege mbiri: Ax + By + CI + D = 0, uye + A¹h V¹u S¹z + + D = 0. Tinogona kutaura kuti vane orthogonal kana Kosi = 0. Izvi zvinoreva kuti NN¹ = AA¹ + VV¹ SS¹ + = 0.

Mukana wokuti vanhu vanga munhu yakafanana ndege

Raireva maviri ndege yakafanana ane hapana pfungwa zvakafanana.

Mamiriro pamusoro inofambirana ndege (equations dzavo vakafanana muzvirevo ndima) ndechokuti vectors N uye N¹, izvo perpendicular kwavari, collinear. Izvi zvinoreva kuti anotevera ezvinhu zvakasangana proportionality:

A / A¹ = B / C = H¹ / S¹.

Kana Proportional okuti zviri kuwedzerwa - A / A¹ = B / C = H¹ / S¹ = DD¹,

ichi chinoratidza kuti umbowo Ndege iyeyo. Izvi zvinoreva kuti mukana wokuti vanhu vanga Ax + By + CI + D = 0 uye + A¹h V¹u S¹z + + D¹ = 0 kutsanangura ndege imwe.

The kure pfungwa ndege

Ngatitii tine ndege P, iyo inopiwa (0). Zvakakosha kuwana chinhambwe kubva pfungwa Coordonatele (hₒ, uₒ, zₒ) = Qₒ. , Unofanira kuunza mukana wokuti vanhu vanga vari mundege II chakaipa chitarisiko kuti:

(Ρ, v) = peji (r≥0).

Muchiitiko chino, ρ (x, y, z) ndiyo nharaunda Vector wedu nguva Q, riri pamusoro N S. - N ndiyo kureba perpendicular, rakabudiswa kubva razero pfungwa, ndima - ndiye chikwata Vector, riri yakarongwa mirayiridzo a.

Musiyano ρ-ρº nharaunda Vector ane pfungwa Q = (x, y, z), kuva N uye nharaunda Vector munhu akapiwa pfungwa Q ₀ = (hₒ, uₒ, zₒ) iri Vector yakadaro, Mhedziso kukosha fungidziro icho pamusoro ndima vakaenzana kure D, izvo zvakafanira kuwana kubva Q = ₀ (hₒ, uₒ, zₒ) kuti P:

D = | (ρ-ρ ^0, v) |, asi

(Ρ-ρ ^0, v) = (ρ, vs. ) - (ρ ^0, v) = S. (ρ ^0, v).

Saka zvinoitika,

D = | (ρ ^0, v) p |.

Zvino zviri pachena kuti kuverenga kure D kubva ₀ kusvika Q ndege P, zvinokosha kushandisa muupenyu maonero ndege mukana wokuti vanhu vanga, chaishanda kune rworuboshwe peji, uye yokupedzisira panzvimbo X, y, z chinotsiva (hₒ, uₒ, zₒ).

Saka, tinowana Mhedziso kukosha zvichizoita mashoko kuti kunodiwa d.

Uchishandisa parameters mutauro, tinowana mashoko pachena:

D = | Ahₒ Vuₒ + + Czₒ | / √ (A² + V² + s²).

Kana wainyatsotaura pfungwa Q ₀ riri mhiri ndege P sezvo mavambo, ipapo pakati Vector ρ-ρ ⁰ uye ndima iri ane obtuse yorusvingo, saka:

D = - (ρ-ρ ^0, v) = (ρ ^0, v) -p> 0.

In Kana apo pfungwa Q ₀ pavanoparidza mavambo iri chete mhiri U, kuti rikwanise angled akasikwa, ndiko:

D = (ρ-ρ ^0, v) = peji - (ρ ^0, v)> 0.

Mugumisiro ndechokuti mune yaimbova nyaya (ρ ^0, v)> s, ari wechipiri (ρ ^0, v)

Uye tangent Ndege kwayo mukana wokuti vanhu vanga

Achitaura nendege pemvura kutoda tangency Mº - ndege ine zvose zvinogoneka tangent kuti munyonga wakavhomorwa kuburikidza iyoyo pamusoro pemvura.

With ichi pose chimiro mukana wokuti vanhu vanga F (x, y, z) = 0 mune mukana wokuti vanhu vanga vari tangent Ndege tangent pfungwa Mº (hº, uº, zº) aizova:

F _X (hº, uº, zº) (hº x) + F _X (hº, uº, zº) (uº Y) + F _X (hº, uº, zº) (Z-zº) = 0.

Kana pamusoro unoiswa zvakajeka Z = d (x, y), ipapo tangent Ndege kunorondedzerwa mukana wokuti vanhu vanga:

Z-zº = d (hº, uº) (hº x) + F (hº, uº) (and uº).

The mharadzano kuti ndege mbiri

In -dimensional matatu nzvimbo iri batanidza system (rectangular) Oxyz, akapiwa maviri ndege P 'uye P' kuti bindepinde uye murege -enderana. Sezvo chero ndege, riri rimwe rectangular batanidza ezvinhu kunorondedzerwa mukuru mukana wokuti vanhu vanga, taifunga kuti n 'uye n "anotsanangurwa nomunyori equations A'x + V'u S'z + + D' = 0 uye A" + B x '+ Y With "z + D" = 0. Muchiitiko ichi tine vakakwana n '(A', B ', C') ndege P 'uye inowanzotorwa n "(A", B ", C") ndege P'. Sezvo ndege yedu havana kuenderana uye regai -enderana, ipapo vectors izvi havana collinear. Tichishandisa mutauro masvomhu, tine ichi chinogona kunyorwa se: n '≠ n "↔ (A', B ', C') ≠ (λ * Uye", λ * In ", λ * C"), λεR. Ngaatorerwe yakarurama mutsetse riri panguva mharadzano P 'uye P ", achaiswa runotaurwa tsamba imwe, mune ichi kana = P' ∩ P".

uye - mutsetse ina muzvizhinji pfungwa (nevakawanda) ndege P 'uye P ". Izvi zvinoreva kuti Coordonatele upi pfungwa kungova mutsetse mumwe, anofanira panguva kugutsa mukana wokuti vanhu vanga A'x + V'u S'z + + D '= 0 uye A "X + B' + C and" Z + D "= 0. Izvi zvinoreva kuti Coordonatele yepfungwa achava imwe mhinduro inotevera equations:

Mugumisiro ndechokuti mhinduro (chikuru) ino equations achasarudza kuti Coordonatele nerimwe pfungwa pamutsetse iyo achaita sezvo pfungwa mharadzano P 'uye P ", uye kuona mutsetse ari batanidza mugadziriro Oxyz (rectangular) nzvimbo.