Ad argvmentvm transi
Topos Uranos
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Vectores in spatio et prodvctvm scalare

Svmmarivm

Haec lectio vectores, qvos lectio de vectoribvs in plano dvabvs coordinatis tractavit, in spativm trivm dimensionvm transfert. Systema trivm axivm perpendicvlarivm orientatione positiva figit, facta geometriae Evclideae spatii qvae admittvntvr declarat et, theoremate Pythagorae bis adhibito, distantiam inter dvo pvncta obtinet. Vectores vt terniones nvmerorvm realivm definit, cvm eorvm svmma, prodvcto per scalarem, basi canonica et vnicitate coordinatarvm; normam et prodvctvm scalare definit, eorvm proprietates demonstrat, inaeqvalitatem Cavchy-Schwarzianam per discriminantem et inaeqvalitatem triangvlarem, et per eas angvlvm dvorvm vectorvm definit, qvi cvm angvlo geometriae elementaris congrvit. Deinde orthogonalitatem, proiectionem orthogonalem, cosinvs directores et scriptvram prodvcti scalaris vt prodvcti versvs per colvmnam tractat, ex qva seqvitvr matrices qvarvm colvmnae svnt vectores vnitarii inter se perpendicvlares longitvdines et angvlos servare. Applicationes svnt opvs vis, theorema cosinvvm in spatio et angvlvs molecvlae methani; problemata de diagonalibvs cvbi et parallelepipedi agvnt, de vectoribvs dvobvs datis perpendicvlaribvs, de tensionibvs trivm fvnivm et de tetraedris qvorvm acies oppositae perpendicvlares svnt.

Proposita discendi

  1. Pvncta in systemate coordinatarvm Cartesianarvm spatii collocare, eorvm octantem agnoscere et distantias compvtare, formvla per theorema Pythagorae confirmata.
  2. Cvm vectoribvs spatii operari, eos in basi canonica exprimere et per eos proprietates parallelogrammorvm, parallelepipedorvm et tetraedrorvm demonstrare.
  3. Normas, prodvcta scalaria, angvlos, proiectiones orthogonales et cosinvs directores compvtare, atqve inaeqvalitates Cavchy-Schwarzianam et triangvlarem in spatio demonstrare.
  4. Prodvctvm scalare vt prodvctvm matricvm scribere et per id decernere vtrvm matrix longitvdines et angvlos servet.
  5. Prodvcto scalari opvs vis, aeqvilibrivm virivm concvrrentivm et angvlos molecvlae exprimere.

Systema coordinatarvm spatii

In lectione cvrsvs Algebra et Geometria I de vectoribvs in plano, vnvmqvodqve pvnctvm plani cvm pari nvmerorvm et qvaeqve translatio cvm alio pari identificata est, et geometria rectarvm, angvlorvm et distantiarvm in compvtationes cvm his paribvs conversa est. Mvndvs physicvs tamen tres dimensiones habet: positio aeroplani, atomi in molecvla vel lampadis e lacvnari pendentis tres nvmeros postvlat. Qvod si ita est, constrvctionem cvm ternionibvs repetere convenit. Pleraqve eorvm qvae in plano demonstrata svnt sine mvtatione extendvntvr, qvia demonstrationes per singvlas componentes procedebant vel solas proprietates algebraicas prodvcti scalaris adhibebant; in his casibvs haec lectio ad lectionem de plano remittit. Qvod cvram postvlat, geometria est: in spatio figvra plana qvae omnia contineat non habetvr, et angvlvs dvorvm vectorvm, exempli gratia, definiri debet anteqvam compvtetvr.

Facta geometriae Evclideae spatii qvae admittvntvr

Geometria analytica spatii geometriam solidorvm convertit, qvam Evclides in libris XI ad XIII Elementorvm exposvit. Eivs constrvctio axiomatica hvnc cvrsvm excedit, vt accidit constrvctioni plani in lectione de aeqvatione rectae, vbi facta geometriae planae qvae admittvntvr in indicem collecta svnt. Eodem modo facta spatii qvibvs haec pars vtitvr sine demonstratione admittvntvr et in indicem infra scriptvm colligvntvr; qvotiens lectio hvivs partis ad vnvm eorvm recvrrit, ad hvnc indicem remittit.

  1. Plana. Per tria pvncta non collinearia vnicvm planvm transit, et si dvo pvncta rectae in plano svnt, tota recta in eo est. In vnoqvoqve plano spatii valent facta geometriae planae in lectione de aeqvatione rectae admissa; nominatim theorema Pythagorae eivsqve conversa, et libertas systematis axivm in eo eligendi.
  2. Rectae et plana perpendicvlaria. Per vnvmqvodqve pvnctvm vnicvm planvm transit rectae datae perpendicvlare, et hoc planvm rectam in vnico pvncto secat. Recta plano perpendicvlaris perpendicvlaris est omnibvs rectis plani qvae per pvnctvm transevnt in qvo planvm secat.
  3. Axes. Tres rectae qvae per pvnctvm OO transevnt et binae perpendicvlares svnt, cvm sensv positivo in vnaqvaqve et eadem mensvra longitvdinis, systema axivm efficivnt. Distantiae, angvli et figvrae ab hac electione non pendent.
  4. Coordinatae. Systemate axivm fixo, plana tribvs axibvs perpendicvlaria per pvnctvm PP dvcta eos in pvnctis abscissarvm xx, yy et zz secant, et vicissim, pro vnoqvoqve ternione nvmerorvm haec tria plana in vnico pvncto se secant. Pvncta qvorvm tertia coordinata aeqvalis est cc planvm efficivnt axi zz perpendicvlare, in qvo dvae primae coordinatae svnt coordinatae Cartesianae plani eadem mensvra; pvncta qvorvm primae coordinatae fixae svnt rectam efficivnt axi zz parallelam, illi plano perpendicvlarem, in qva tertia coordinata positionem eadem mensvra metitvr. Idem valet axibvs permvtatis.

Qvartvm factvm est consectarivm trivm primorvm, qvod geometria solidorvm per proprietates rectarvm parallelarvm et planorvm parallelorvm demonstrat; separatim envntiatvr qvia ea est forma qva adhibebitvr.

Systema coordinatarvm Cartesianarvm spatii est systema axivm, qvae axis xx, axis yy et axis zz appellantvr et in origine OO se secant. Vnvmqvodqve pvnctvm PP cvm ternione svarvm coordinatarvm identificatvr, P=(x,y,z)P = (x, y, z), qvae abscissa, ordinata et applicata appellantvr. Tria plana coordinata svnt planvm xyxy, aeqvationis z=0z = 0; planvm yzyz, aeqvationis x=0x = 0; et planvm xzxz, aeqvationis y=0y = 0.

Plana coordinata spativm in octo regiones dividvnt, qvae octantes appellantvr, secvndvm signvm cvivsqve coordinatae; primvs octans est pvnctorvm qvorvm tres coordinatae positivae svnt, et ceteri ternione signorvm svorvm nominantvr, exempli gratia octans (+,−,+)(+, -, +). In plano dvo axes perpendicvlares dvobvs modis essentialiter diversis disponi possvnt: secvndvs axis obtinetvr primo per qvadrantem conversionis contra motvm horologii verso, avt secvndvm motvm horologii. In spatio idem cvm tribvs axibvs accidit, et distinctio nomen habet.

Systema coordinatarvm spatii orientationem positivam habet cvm, a latere positivo axis zz inspectvm, qvadrans conversionis qvi semiaxem positivvm xx in semiaxem positivvm yy dvcit contra motvm horologii videtvr. Aeqvivalenter valet regvla manvs dextrae: si digiti manvs dextrae a semiaxe positivo xx ad semiaxem positivvm yy cvrvantvr, pollex semiaxem positivvm zz monstrat.

Si sensvs vnivs axivm invertitvr, orientatio mvtatvr; si dvorvm, servatvr. Axibvs circvlariter permvtatis, a xx, yy, zz ad yy, zz, xx, orientatio ne sic qvidem mvtatvr, ita vt in systemate positivo, a semiaxe positivo xx inspecto, conversio a yy ad zz qvoqve contra motvm horologii fiat. Posthac omnia systemata orientationem positivam habent. Figvrae ea in perspectiva eqvestri delineant: axem yy dextrorsvm, axem zz svrsvm et axem xx ad spectatorem, obliqve dvctvm et longitvdinibvs ad dimidivm redactis. Orientatio in hac lectione nihil confert; in proxima lectione, de prodvcto vectoriali et prodvcto mixto, essentialis erit.

ExemplvmPvnctvm in spatio collocare

Pvnctvm P=(3,4,2)P = (3, 4, 2) per axes procedendo collocare, eivs octantem indicare et pvnctvm P1=(3,4,−2)P_1 = (3, 4, -2) collocare.

Demonstratio

  1. O=(0,0,0)O = \left( 0, 0, 0 \right)

    Tres axes orientatione positiva delineamvs: desvper inspectvs, qvadrans conversionis a semiaxe positivo xx ad semiaxem positivvm yy contra motvm horologii fit.

  2. A=(3,0,0)A = \left( \resaltar{3}, 0, 0 \right)

    Spativm longitvdinis 33 per axem xx procedimvs.

  3. B=(3,4,0)B = \left( 3, \resaltar{4}, 0 \right)

    Ab AA spativm longitvdinis 44 in directione axis yy procedimvs, e plano xyxy non egressi.

  4. P=(3,4,2)P = \left( 3, 4, \resaltar{2} \right)

    Ab BB spativm longitvdinis 22 in directione axis zz ascendimvs. Tres coordinatae positivae svnt: PP in primo octante est, et vertex est oppositvs ipsi OO capsae cvivs acies svnt 33, 44 et 22.

  5. P1=(3,4,−2)P_1 = \left( 3, 4, \resaltar{-2} \right)

    Si ab BB spativm longitvdinis 22 descenditvr, ad P1P_1 pervenitvr, in octante (+,+,−)(+, +, -). Planvm xyxy segmentvm PP1P P_1 perpendicvlariter in eivs pvncto medio BB secat: P1P_1 est pvnctvm symmetricvm ipsivs PP respectv illivs plani.

Distantia inter dvo pvncta

In plano distantia inter dvo pvncta obtenta est theoremate Pythagorae semel adhibito ad triangvlvm cvivs catheti axibvs paralleli svnt. In spatio bis adhibendvm est: primvm in plano horizontali, deinde in plano verticali.

TheoremaDistantia inter dvo pvncta

Distantia inter pvncta P=(x1,y1,z1)P = (x_1, y_1, z_1) et Q=(x2,y2,z2)Q = (x_2, y_2, z_2) formvla infra scripta datvr.

d⁡(P,Q)=(x2−x1)2+(y2−y1)2+(z2−z1)2\operatorname{d}(P, Q) = \sqrt{\left( x_2 - x_1 \right)^2 + \left( y_2 - y_1 \right)^2 + \left( z_2 - z_1 \right)^2}

Pvnctvm avxiliare RR introdvcitvr, in altitvdine ipsivs PP et in verticali ipsivs QQ. Figvra adhibet P=(1,2,1)P = (1, 2, 1) et Q=(3,5,7)Q = (3, 5, 7).

Demonstratio

  1. R=(x2,y2,z1)R = \left( x_2, y_2, \resaltar{z_1} \right)

    Sit RR pvnctvm cvm dvabvs primis coordinatis ipsivs QQ et applicata ipsivs PP. Pvncta PP et RR in plano applicatae z1z_1 svnt, et RR et QQ in recta axi zz parallela, qvae ex pvnctis primarvm coordinatarvm x2x_2 et y2y_2 constat.

  2. d⁡(P,R)2=(x2−x1)2+(y2−y1)2{\operatorname{d}(P, R)}^2 = \dato{pr}{\left( x_2 - x_1 \right)^2 + \left( y_2 - y_1 \right)^2}

    Per factvm de coordinatis, in plano applicatae z1z_1 dvae primae coordinatae Cartesianae svnt, et formvla distantiae in plano, ex lectione de aeqvatione rectae, longitvdinem ipsivs PRPR dat. In figvra, 4+9=134 + 9 = 13.

  3. d⁡(R,Q)=∣z2−z1∣\operatorname{d}(R, Q) = \dato{rq}{\abs{z_2 - z_1}}

    In recta verticali qvae RR et QQ continet, applicata positionem metitvr: longitvdo ipsivs RQRQ est valor absolvtvs differentiae applicatarvm. In figvra, 7−1=67 - 1 = 6.

  4. d⁡(P,Q)2=d⁡(P,R)2+d⁡(R,Q)2{\operatorname{d}(P, Q)}^2 = \resaltar{{\operatorname{d}(P, R)}^2} + \resaltar{{\operatorname{d}(R, Q)}^2}

    Recta RQRQ perpendicvlaris est plano applicatae z1z_1 et, per secvndvm factvm admissvm, rectae PRPR, qvae in illo plano est et per RR transit. Triangvlvm PRQPRQ rectangvlvm est in RR et, per primvm factvm, theorema Pythagorae valet in plano qvod id continet.

  5. d⁡(P,Q)=(x2−x1)2+(y2−y1)2+(z2−z1)2,13+36=7\operatorname{d}(P, Q) = \sqrt{\resaltar{\left( x_2 - x_1 \right)^2 + \left( y_2 - y_1 \right)^2} + \resaltar{\left( z_2 - z_1 \right)^2}}, \qquad \sqrt{13 + 36} = 7

    Svbstitvimvs et radicem qvadratam extrahimvs. Si P=RP = R vel R=QR = Q, triangvlvm non est, sed formvla ad formvlam vnivs tantvm distantiarvm redigitvr et vera manet. In figvra distantia est 77.

Speciatim distantia ipsivs P=(x,y,z)P = (x, y, z) ab origine est x2+y2+z2\sqrt{x^2 + y^2 + z^2}, longitvdo diagonalis capsae cvivs acies svnt ∣x∣\abs{x}, ∣y∣\abs{y} et ∣z∣\abs{z}. Exempli gratia, diagonalis cvbi aciei aa metitvr a2+a2+a2=a3\sqrt{a^2 + a^2 + a^2} = a \sqrt{3}.