Geometria analytica spatii: synthesis et provocationes

Svmmarivm

Haec lectio partem de geometria analytica spatii clavdit, qvae cvrsvs vltima est. In catenam dedvctivam colligit qvae in qvattvor eivs lectionibvs argvmenti consecvta svnt: distantiam inter dvo pvncta, theoremate Pythagorae bis adhibito obtentam; prodvctvm scalare, inaeqvalitatem Cavchy-Schwarzianam et angvlvm dvorvm vectorvm, qvi cvm angvlo geometriae elementaris congrvit; prodvctvm vectoriale et prodvctvm mixtvm, ex determinante ordinis tertii orta, cvm eorvm interpretatione vt area, vt volvmen et vt criterivm complanaritatis; aeqvationes rectarvm et planorvm et classificationem positionvm mvtvarvm per theorema Rovché-Frobenii; ac deniqve distantias, angvlos, proiectiones et sphaeram. Explicat qvid pars a partibvs antecedentibvs et a cvrsv Algebra et Geometria I accipiat, qvid sine demonstratione admissvm sit et qvo dvcat. In probatione finali facvltates probationis initialis introdvctionis repetit, et decem provocationibvs finitvr, difficilioribvs qvam problemata lectionvm, qvae plvres earvm conivngvnt.

Proposita discendi

Catenam definitionvm et theorematvm partis restitvere, a distantia inter dvo pvncta vsqve ad perpendicvlarem commvnem et sphaeram, in qvoqve anvlo indicando qvibvs effectis antecedentibvs eivs demonstratio vtatvr.
In parte demonstrata ab admissis distingvere, eivsqve effecta cvm determinantibvs et systematibvs partis de matricibvs et cvm geometria analytica plani cvrsvs I conivngere.
Problemata solvere qvae prodvcta scalare, vectoriale et mixtvm cvm aeqvationibvs rectarvm, planorvm et sphaerarvm conivngvnt, in qvoqve casv instrvmentvm et systema axivm aptissima eligendo.
Geometria spatii res crystallographiae, delineationis technicae, opticae et navigationis ad formam mathematicam redigere, et solvtionem cvm sva positione, svo calcvlo et svo examine verisimilitvdinis commvnicare.

Qvae vnitas consecvta est

Coordinatae et prodvctvm scalare

Pars incepit declarando qvid demonstratvra non esset. Lectio de vectoribvs in spatio et prodvcto scalari facta geometriae Evclideae spatii qvae admittvntvr in indicem collegit et, theoremate Pythagorae bis adhibito, formvlam distantiae obtinvit. Vectores vt terniones definiti svnt, et prodvctvm scalare proprietates prodvcti scalaris plani servavit, cvm inaeqvalitate Cavchy-Schwarziana et inaeqvalitate triangvlari. Atqvi in spatio nvlla figvra plana est qvae omnia contineat, et ordo inversvs est: angvlvs per svvm cosinvm definitvs est, qvem inaeqvalitas Cavchy-Schwarziana inter $-1$ et $1$ collocat, et deinde demonstratvm est evm metiri qvod geometria elementaris angvlvm vocat, qvia prodvctvm scalare a solis longitvdinibvs latervm triangvli pendet. Ex qvo secvta svnt proiectio orthogonalis, cosinvs directores et matrices qvae longitvdines et angvlos servant.

Prodvctvm vectoriale et prodvctvm mixtvm

Qvaestio de vectore ad dvos vectores datos perpendicvlari in lectione de prodvcto vectoriali et prodvcto mixto ad tria determinantia ordinis secvndi dvxit. Qvod si ita est, proprietates determinantivm statim anticommvtativitatem et bilinearitatem dedervnt; identitas Lagrangiana normam vt aream parallelogrammi dedit; et formvla dvplicis prodvcti vectorialis, qvae associativitatis locvm tenet, ostendit vectores ad dvos vectores non parallelos perpendicvlares esse mvltiplicia eorvm prodvcti vectorialis. Prodvctvm mixtvm determinans ordinis tertii esse apparvit: evanescit si et solvm si tres vectores complanares svnt, eivs valor absolvtvs est volvmen parallelepipedi et eivs qvadratvm est determinans Gramianvm. Lectio deniqve separavit qvod de sensv prodvcti vectorialis demonstratvr ab eo qvod in axibvs delineandis convenitvr, systemate dextrorso, et ab eo qvod admittitvr, regvla manvs dextrae.

Rectae et plana

Lectio de rectis et planis in spatio pertinentiam ad planvm ad vnam condicionem redegit, $\vec{n} \cdot \vec{AP} = 0$ , vbi $\vec{n}$ est prodvctvm vectoriale vectorvm directionis, et ex ea aeqvationem Cartesianam obtinvit, vnicam praeter factorem. Qvo facto, pvncta plvribvs planis commvnia svnt solvtiones systematis linearis, et theorema Rovché-Frobenii svam interpretationem geometricam accepit: solvtiones systematis compatibilis in tribvs incognitis pvnctvm, rectam vel planvm efficivnt pro rango. Ex ea dedvctae svnt positiones mvtvae planorvm et rectarvm, cvm prodvcto mixto $\vec{AB} \cdot \left( \vec{u} \times \vec{v} \right)$ vt criterio ad rectas secantes a rectis non coplanaribvs distingvendas, necnon fascis planorvm.

Mensvrae

Lectio de distantiis, angvlis et volvminibvs in spatio tria prodvcta cvm aeqvationibvs conivnxit. Proiciendo in normalem vel in rectam, formvlas distantiae pvncti a plano et a recta demonstravit, pede perpendicvlaris vt vnico pvncto proximo, et theorema centrale partis: dvae rectae non coplanares vnicam perpendicvlarem commvnem habent, cvivs longitvdo est prodvctvm mixtvm per normam divisvm. Sic solvitvr qvaestio de dvobvs cvnicvlis ferriviae svbterraneae qvam introdvctio posvit. Deinde demonstravit angvlos inter rectas et plana per prodvcta scalaria definitos cvm angvlis geometriae elementaris congrvere, aream proiectionis polygoni plani esse eivs aream cosinv angvli dihedralis mvltiplicatam, et sphaeram eivsqve planvm tangens investigavit.

Itaqve pars catenam efficit: facta admissa distantiam dant; distantia et proprietates prodvcti scalaris angvlvm dant; determinantia prodvctvm vectoriale et prodvctvm mixtvm dant; haec pertinentiam ad planvm in aeqvationem linearem convertvnt; systemata linearia positiones ordinant; et proiectio orthogonalis, illis aeqvationibvs adhibita, metitvr. Pictvra qvae seqvitvr qvattvor segmenta in vnam catenam colligit. Qvisqve nodvs colvmnae est definitio avt theorema, secvndvm ordinem logicvm nvmeratvs et arcv cvm anvlis antecedentibvs conivnctvs qvibvs eivs demonstratio vtitvr; circvm flvitant fvlcra, id est principia et instrvmenta ad qvae catena itervm atqve itervm recvrrit, qvodqve linea continva cvm anvlis qvi eo egent et linea pvnctata cvm anvlo in qvo stabilitvr conivnctvm, praeter ea qvae sine demonstratione admittvntvr avt ex partibvs antecedentibvs vel ex cvrsv I venivnt, qvae vt circvli vacvi pingvntvr. Fvlcro pvlsato omnes anvli qvi eo vtvntvr illvstrantvr, et qvisqve anvlvs ad lectionem remittit in qva demonstratvr.

Fvlcra catenae

AFacta Evclidea spatii. Admittvntvr exsistentia et vnicitas plani per tria pvncta non collinearia et plani ad rectam perpendicvlaris per pvnctvm, libertas axes eligendi et correspondentia inter pvncta et terniones. In hac vnitate sine demonstratione admittitvr.
BAreae et volvmina elementaria. Admittvntvr basis in altitvdinem pro parallelogrammo et parallelepipedo, dimidivm hvivs areae pro triangvlo, tertia portio basis in altitvdinem pro pyramide et additivitas areae; eorvm fvndamentvm est calcvlvs integralis. In hac vnitate sine demonstratione admittitvr.
CGeometria plani. Distantia, proprietates prodvcti scalaris, angvlvs dvorvm vectorvm, lemma de vectoribvs aeqvalibvs et parallelogrammis, qvadrati completio et signvm polynomii qvadratici pro eivs discriminante, in cvrsv I demonstrata. In hac vnitate sine demonstratione admittitvr.
DTrigonometria. Arcvs cosinvs vt biiectio ex $\icc{-1}{1}$ in $\icc{0}{\pi}$ , identitas $\cos^2 \theta + \sin^2 \theta = 1$ et theorema cosinvvm, in partibvs 1 et 2 demonstrata. In hac vnitate sine demonstratione admittitvr.
EMatrices et determinantia. Transposita prodvcti, inversa vnilateralis, proprietates determinantis, regvla Crameri et aeqvivalentia inter $\det A \neq 0$ et rangvm maximvm, in parte 4 demonstratae. In hac vnitate sine demonstratione admittitvr.
FRovché-Frobenivs. Systema $AX = B$ compatibile est si et solvm si $\operatorname{rg}(A) = \operatorname{rg}(M)$ , et tvm tot incognitas liberas habet qvot nvmervs incognitarvm minvs rangvs; in parte 4 demonstratvm. In hac vnitate sine demonstratione admittitvr.
GProiectio orthogonalis. Si $\vec{v}$ nvllvs non est, omnis vector vnico modo scribitvr vt mvltiplex ipsivs $\vec{v}$ plvs vector ad $\vec{v}$ orthogonalis, et qvadratvm eivs normae est svmma qvadratorvm normarvm vtrivsqve componentis. Constitvitvr in: 6. Proiectio orthogonalis.
HPerpendicvlares ad dvos vectores. Si $\vec{u}$ et $\vec{v}$ neqve nvlli neqve paralleli svnt, vectores ad vtrvmqve perpendicvlares svnt prorsvs mvltiplicia ipsivs $\vec{u} \times \vec{v}$ . Constitvitvr in: 14. Perpendicvlares ad dvos vectores.

Segmentvm 1 Coordinatae et prodvctvm scalare
- Pvncta in spatio collocare et distantias compvtare, formvlam theoremate Pythagorae confirmando.
- Cvm vectoribvs spatii operari et per eos proprietates parallelogrammorvm et parallelepipedorvm demonstrare.
- Inaeqvalitatem Cavchy-Schwarzianam et inaeqvalitatem triangvlarem demonstrare, et per eas angvlvm dvorvm vectorvm definire.
- Proiectiones orthogonales et cosinvs directores compvtare, et matrices qvae longitvdines et angvlos servant agnoscere.
1. Distantia inter dvo pvncta. Distantia inter $(x_1, y_1, z_1)$ et $(x_2, y_2, z_2)$ est $\sqrt{\left( x_2 - x_1 \right)^2 + \left( y_2 - y_1 \right)^2 + \left( z_2 - z_1 \right)^2}$ , per theorema Pythagorae in plano horizontali et deinde in verticali adhibitvm.
  Nititvr: A (Facta Evclidea spatii), C (Geometria plani). Demonstratvr in: Vectores in spatio et prodvctvm scalare.
2. Vectores et basis canonica. Terniones spativm vectoriale efficivnt, omnis vector vnico modo est combinatio vectorvm $\vec{i}$ , $\vec{j}$ et $\vec{k}$ , et si $\vec{PQ} = \vec{RS}$ et qvattvor pvncta non svnt collinearia, in plano iacent et $PQSR$ est parallelogrammvm, et vicissim.
  Vtitvr: 1. Distantia inter dvo pvncta. Nititvr: A (Facta Evclidea spatii), C (Geometria plani). Demonstratvr in: Vectores in spatio et prodvctvm scalare.
3. Proprietates prodvcti scalaris. Prodvctvm scalare commvtativvm et distribvtivvm est, nvmeros extrahi sinit et $\vec{u} \cdot \vec{u} = {\norma{\vec{u}}}^2$ satisfacit; qvam ob rem ${\norma{\vec{u} \pm \vec{v}}}^2 = {\norma{\vec{u}}}^2 \pm 2 \vec{u} \cdot \vec{v} + {\norma{\vec{v}}}^2$ .
  Vtitvr: 2. Vectores et basis canonica. Nititvr: C (Geometria plani). Demonstratvr in: Vectores in spatio et prodvctvm scalare.
4. Cavchy-Schwarz et inaeqvalitas triangvlaris. Valent $\abs{\vec{u} \cdot \vec{v}} \le \norma{\vec{u}} \norma{\vec{v}}$ et $\norma{\vec{u} + \vec{v}} \le \norma{\vec{u}} + \norma{\vec{v}}$ , cvm aeqvalitate in prima si et solvm si alter vector mvltiplex alterivs est.
  Vtitvr: 3. Proprietates prodvcti scalaris. Nititvr: C (Geometria plani). Demonstratvr in: Vectores in spatio et prodvctvm scalare.
5. Angvlvs est geometricvs. Nvmervs $\theta$ ex $\icc{0}{\pi}$ , pro qvo $\cos \theta = \frac{\vec{u} \cdot \vec{v}}{\norma{\vec{u}} \norma{\vec{v}}}$ , est mensvra angvli qvem sagittae vectorvm $\vec{u}$ et $\vec{v}$ ab eadem origine dvctae efficivnt; praesertim dvo vectores non nvlli perpendicvlares svnt si et solvm si eorvm prodvctvm scalare nvllvm est.
  Vtitvr: 1. Distantia inter dvo pvncta, 3. Proprietates prodvcti scalaris, 4. Cavchy-Schwarz et inaeqvalitas triangvlaris. Nititvr: A (Facta Evclidea spatii), C (Geometria plani), D (Trigonometria). Demonstratvr in: Vectores in spatio et prodvctvm scalare.
6. Proiectio orthogonalis. Si $\vec{v} \neq \vec{0}$ , omnis $\vec{u}$ vnico modo est $\vec{p} + \vec{w}$ , cvm $\vec{p} = \frac{\vec{u} \cdot \vec{v}}{{\norma{\vec{v}}}^2} \vec{v}$ et $\vec{w}$ ad $\vec{v}$ orthogonali, et ${\norma{\vec{u}}}^2 = {\norma{\vec{p}}}^2 + {\norma{\vec{w}}}^2$ .
  Vtitvr: 3. Proprietates prodvcti scalaris, 5. Angvlvs est geometricvs. Demonstratvr in: Vectores in spatio et prodvctvm scalare.
7. Matrices orthogonales. Valet $\left( A \vec{u} \right) \cdot \vec{v} = \vec{u} \cdot \left( A^{\mathsf{T}} \vec{v} \right)$ ; si colvmnae matricis $A$ vnitariae et orthogonales svnt, $A^{\mathsf{T}} A = I$ et $A$ normas, distantias et angvlos servat.
  Vtitvr: 3. Proprietates prodvcti scalaris. Nititvr: E (Matrices et determinantia). Demonstratvr in: Vectores in spatio et prodvctvm scalare.
Segmentvm 2 Prodvctvm vectoriale et prodvctvm mixtvm
- Prodvctvm vectoriale per determinans formale definire et per determinantia demonstrare id anticommvtativvm et bilineare esse, non avtem associativvm.
- Perpendicvlaritatem, identitatem Lagrangianam, criterivm parallelismi et formvlam dvplicis prodvcti vectorialis demonstrare.
- Prodvctvm mixtvm cvm determinante ordinis tertii identificare et id vt volvmen et vt criterivm complanaritatis interpretari.
- In sensv prodvcti vectorialis qvod demonstratvr a conventione systematis dextrorsi et a regvla manvs dextrae, qvae admittitvr, separare.
1. Proprietates algebraicae. Prodvctvm vectoriale, cvivs componentes svnt tria determinantia ordinis secvndi, anticommvtativvm et bilineare est, $\vec{u} \times \vec{u} = \vec{0}$ satisfacit neqve associativvm est.
  Nititvr: E (Matrices et determinantia). Demonstratvr in: Prodvctvm vectoriale et prodvctvm mixtvm.
2. Perpendicvlaritas. Valent $\vec{u} \cdot \left( \vec{u} \times \vec{v} \right) = 0$ et $\vec{v} \cdot \left( \vec{u} \times \vec{v} \right) = 0$ : prodvctvm vectoriale ad ambos svos factores perpendicvlare est.
  Vtitvr: 3. Proprietates prodvcti scalaris, 8. Proprietates algebraicae. Demonstratvr in: Prodvctvm vectoriale et prodvctvm mixtvm.
3. Identitas Lagrangiana. Pro omnibvs vectoribvs valet ${\norma{\vec{u} \times \vec{v}}}^2 = {\norma{\vec{u}}}^2 {\norma{\vec{v}}}^2 - \left( \vec{u} \cdot \vec{v} \right)^2$ .
  Vtitvr: 3. Proprietates prodvcti scalaris, 8. Proprietates algebraicae. Demonstratvr in: Prodvctvm vectoriale et prodvctvm mixtvm.
4. Area parallelogrammi. Norma $\norma{\vec{u} \times \vec{v}} = \norma{\vec{u}} \norma{\vec{v}} \sin \theta$ est area parallelogrammi latervm $\vec{u}$ et $\vec{v}$ , et triangvlvm $ABC$ aream $\frac{1}{2} \norma{\vec{AB} \times \vec{AC}}$ habet.
  Vtitvr: 5. Angvlvs est geometricvs, 10. Identitas Lagrangiana. Nititvr: B (Areae et volvmina elementaria), D (Trigonometria), G (Proiectio orthogonalis). Demonstratvr in: Prodvctvm vectoriale et prodvctvm mixtvm.
5. Prodvctvm nvllvm et parallelismvs. Valet $\vec{u} \times \vec{v} = \vec{0}$ si et solvm si alter e dvobvs vectoribvs mvltiplex alterivs est.
  Vtitvr: 8. Proprietates algebraicae. Demonstratvr in: Prodvctvm vectoriale et prodvctvm mixtvm.
6. Dvplex prodvctvm vectoriale. Valent $\vec{u} \times \left( \vec{v} \times \vec{w} \right) = \left( \vec{u} \cdot \vec{w} \right) \vec{v} - \left( \vec{u} \cdot \vec{v} \right) \vec{w}$ et identitas Iacobiana.
  Vtitvr: 3. Proprietates prodvcti scalaris, 8. Proprietates algebraicae. Demonstratvr in: Prodvctvm vectoriale et prodvctvm mixtvm.
7. Perpendicvlares ad dvos vectores. Si $\vec{u}$ et $\vec{v}$ neqve nvlli neqve paralleli svnt, vector ad vtrvmqve perpendicvlaris est si et solvm si mvltiplex ipsivs $\vec{n} = \vec{u} \times \vec{v}$ est, et tvm est $\frac{\vec{x} \cdot \vec{n}}{{\norma{\vec{n}}}^2} \vec{n}$ .
  Vtitvr: 9. Perpendicvlaritas, 12. Prodvctvm nvllvm et parallelismvs, 13. Dvplex prodvctvm vectoriale. Demonstratvr in: Prodvctvm vectoriale et prodvctvm mixtvm.
8. Prodvctvm mixtvm est determinans. Prodvctvm mixtvm $\vec{u} \cdot \left( \vec{v} \times \vec{w} \right)$ est determinans matricis cvivs ordines svnt $\vec{u}$ , $\vec{v}$ et $\vec{w}$ ; non mvtatvr factoribvs circvlariter permvtatis, signvm mvtat dvobvs inter se commvtatis, et pvnctvm et crvx inter se commvtari possvnt.
  Vtitvr: 3. Proprietates prodvcti scalaris, 8. Proprietates algebraicae. Nititvr: E (Matrices et determinantia). Demonstratvr in: Prodvctvm vectoriale et prodvctvm mixtvm.
9. Criterivm complanaritatis. Tres vectores complanares svnt, id est combinationem nvllam cvm aliqvo coefficiente non nvllo admittvnt, si et solvm si eorvm prodvctvm mixtvm nvllvm est.
  Vtitvr: 15. Prodvctvm mixtvm est determinans. Nititvr: E (Matrices et determinantia). Demonstratvr in: Prodvctvm vectoriale et prodvctvm mixtvm.
10. Volvmen et matrix Gramiana. Parallelepipedvm aciervm $\vec{u}$ , $\vec{v}$ et $\vec{w}$ volvmen $\abs{\vec{u} \cdot \left( \vec{v} \times \vec{w} \right)}$ habet, tetraedrvm sextam eivs portionem, et qvadratvm prodvcti mixti est determinans Gramianvm.
  Vtitvr: 11. Area parallelogrammi, 12. Prodvctvm nvllvm et parallelismvs, 15. Prodvctvm mixtvm est determinans, 16. Criterivm complanaritatis. Nititvr: B (Areae et volvmina elementaria), G (Proiectio orthogonalis), E (Matrices et determinantia). Demonstratvr in: Prodvctvm vectoriale et prodvctvm mixtvm.
11. Characterismvs et orientatio. Si $\vec{u}$ et $\vec{v}$ neqve nvlli neqve paralleli svnt, $\vec{u} \times \vec{v}$ est vnicvs vector ad vtrvmqve perpendicvlaris, normam areae habens et prodvctvm mixtvm positivvm; qvod eo vergat qvo pollex manvs dextrae monstrat admittitvr, delineatione axivm semel conventa.
  Vtitvr: 11. Area parallelogrammi, 15. Prodvctvm mixtvm est determinans. Nititvr: H (Perpendicvlares ad dvos vectores). Demonstratvr in: Prodvctvm vectoriale et prodvctvm mixtvm.
Segmentvm 3 Rectae et plana
- Rectas forma vectoriali, parametrica et symmetrica, et plana forma parametrica et Cartesiana scribere, ab altera ad alteram per prodvctvm vectoriale transevndo.
- Demonstrare omne planvm aeqvationem Cartesianam habere, vnicam praeter factorem, et omnem aeqvationem linearem planvm describere.
- Positiones mvtvas planorvm et rectarvm per rangos systematis linearis et per prodvctvm mixtvm diivdicare.
- Fasce planorvm plana constrvere qvae rectam continent et aliam condicionem satisfacivnt.
1. Recta per dvo pvncta. Per dvo pvncta distincta $A$ et $B$ transit vnica recta, $A + t \vec{AB}$ .
  Vtitvr: 2. Vectores et basis canonica. Demonstratvr in: Rectae et plana in spatio.
2. Combinationes dvorvm vectorvm. Si $\vec{u}$ et $\vec{v}$ neqve nvlli neqve paralleli svnt, $\vec{w}$ est eorvm combinatio si et solvm si $\left( \vec{u} \times \vec{v} \right) \cdot \vec{w} = 0$ , et tvm coefficientes vnici svnt.
  Vtitvr: 9. Perpendicvlaritas, 15. Prodvctvm mixtvm est determinans. Nititvr: F (Rovché-Frobenivs), E (Matrices et determinantia). Demonstratvr in: Rectae et plana in spatio.
3. Aeqvatio Cartesiana plani. Pvnctvm in plano $A + \lambda \vec{u} + \mu \vec{v}$ iacet si et solvm si $\left( \vec{u} \times \vec{v} \right) \cdot \vec{AP} = 0$ ; omne planvm aeqvationem $ax + by + cz = d$ habet, vnicam praeter factorem, et omnis talis aeqvatio planvm describit.
  Vtitvr: 12. Prodvctvm nvllvm et parallelismvs, 20. Combinationes dvorvm vectorvm. Nititvr: H (Perpendicvlares ad dvos vectores). Demonstratvr in: Rectae et plana in spatio.
4. Positiones planorvm. Solvtiones systematis compatibilis in $x$ , $y$ et $z$ pvnctvm, rectam vel planvm efficivnt provt rangvs est $3$ , $2$ vel $1$ ; ex qvo positiones dvorvm et trivm planorvm et eae rectae et plani seqvvntvr.
  Vtitvr: 19. Recta per dvo pvncta, 20. Combinationes dvorvm vectorvm, 21. Aeqvatio Cartesiana plani, 15. Prodvctvm mixtvm est determinans. Nititvr: F (Rovché-Frobenivs), E (Matrices et determinantia), H (Perpendicvlares ad dvos vectores). Demonstratvr in: Rectae et plana in spatio.
5. Positiones dvarvm rectarvm. Dvae rectae $A + t \vec{u}$ et $B + \lambda \vec{v}$ non parallelae secant se si $\vec{AB} \cdot \left( \vec{u} \times \vec{v} \right) = 0$ , alioqvin non coplanares svnt; illvd prodvctvm mixtvm nvllvm est si et solvm si in plano iacent.
  Vtitvr: 19. Recta per dvo pvncta, 15. Prodvctvm mixtvm est determinans, 21. Aeqvatio Cartesiana plani. Nititvr: F (Rovché-Frobenivs), E (Matrices et determinantia), H (Perpendicvlares ad dvos vectores). Demonstratvr in: Rectae et plana in spatio.
6. Fascis planorvm. Plana qvae rectam dvobvs planis secantibvs commvnem continent svnt combinationes eorvm aeqvationvm cvm coefficientibvs non ambobvs nvllis.
  Vtitvr: 20. Combinationes dvorvm vectorvm, 22. Positiones planorvm. Demonstratvr in: Rectae et plana in spatio.
Segmentvm 4 Mensvrae
- Formvlas distantiarvm pvncti a plano et a recta demonstrare, pede perpendicvlaris vt vnico pvncto proximo.
- Perpendicvlarem commvnem dvarvm rectarvm non coplanarivm invenire, demonstrare eam vnicam esse et eivs longitvdinem per prodvctvm mixtvm compvtare.
- Angvlos inter rectas et plana definire, demonstrare eos cvm geometricis congrvere et aream figvrae planae cvm area eivs proiectionis conivngere.
- Sphaeram, eivs intersectionem cvm plano et eivs planvm tangens investigare, et haec omnia ad crystallographiam et ad locationem per satellites adhibere.
1. Distantia a plano. Distantia pvncti $(x_0, y_0, z_0)$ a plano $ax + by + cz = d$ est $\frac{\abs{a x_0 + b y_0 + c z_0 - d}}{\sqrt{a^2 + b^2 + c^2}}$ et solvm in pede perpendicvlaris attingitvr; ex ea semispatia et pvnctvm symmetricvm pvncti orivntvr.
  Vtitvr: 21. Aeqvatio Cartesiana plani. Nititvr: G (Proiectio orthogonalis). Demonstratvr in: Distantiae, angvli et volvmina in spatio.
2. Distantia a recta. Distantia pvncti $P$ a recta $A + t \vec{u}$ est $\frac{\norma{\vec{AP} \times \vec{u}}}{\norma{\vec{u}}}$ et solvm in pede perpendicvlaris attingitvr.
  Vtitvr: 10. Identitas Lagrangiana, 19. Recta per dvo pvncta. Nititvr: G (Proiectio orthogonalis). Demonstratvr in: Distantiae, angvli et volvmina in spatio.
3. Perpendicvlaris commvnis. Dvae rectae non coplanares, $A + t \vec{u}$ et $B + s \vec{v}$ , vnicam perpendicvlarem commvnem habent, et distantia inter eas est $\frac{\abs{\vec{AB} \cdot \left( \vec{u} \times \vec{v} \right)}}{\norma{\vec{u} \times \vec{v}}}$ , qvae solvm inter eivs pedes attingitvr.
  Vtitvr: 10. Identitas Lagrangiana, 15. Prodvctvm mixtvm est determinans, 23. Positiones dvarvm rectarvm. Nititvr: E (Matrices et determinantia), H (Perpendicvlares ad dvos vectores), G (Proiectio orthogonalis). Demonstratvr in: Distantiae, angvli et volvmina in spatio.
4. Angvli cvm planis et dihedrales. Angvlvs rectae cvm plano, cvivs sinvs est $\frac{\abs{\vec{u} \cdot \vec{n}}}{\norma{\vec{u}} \norma{\vec{n}}}$ , is est qvem cvm sva proiectione efficit, et angvlvs inter normales dvorvm planorvm est, praeter svpplementvm, angvlvs dihedralis.
  Vtitvr: 4. Cavchy-Schwarz et inaeqvalitas triangvlaris, 11. Area parallelogrammi, 13. Dvplex prodvctvm vectoriale, 15. Prodvctvm mixtvm est determinans, 25. Distantia a plano. Nititvr: D (Trigonometria), H (Perpendicvlares ad dvos vectores), G (Proiectio orthogonalis). Demonstratvr in: Distantiae, angvli et volvmina in spatio.
5. Area proiectionis. Si polygonvm areae $S$ in plano iacet qvod angvlvm dihedralem $\alpha$ cvm plano $\pi$ efficit, cvm $\alpha = 0$ si parallela svnt, eivs proiectio orthogonalis in $\pi$ aream $S \cos \alpha$ habet.
  Vtitvr: 11. Area parallelogrammi, 15. Prodvctvm mixtvm est determinans, 25. Distantia a plano, 28. Angvli cvm planis et dihedrales. Nititvr: B (Areae et volvmina elementaria), H (Perpendicvlares ad dvos vectores). Demonstratvr in: Distantiae, angvli et volvmina in spatio.
6. Sphaera eivsqve planvm tangens. Planvm distantia $\delta$ a centro sphaerae radii $R$ eam secat in circvlo radii $\sqrt{R^2 - \delta^2}$ si $\delta < R$ et eam solvm in pede perpendicvlaris tangit si $\delta = R$ ; planvm tangens in $T$ normalem $\vec{CT}$ habet.
  Vtitvr: 1. Distantia inter dvo pvncta, 25. Distantia a plano. Nititvr: C (Geometria plani), G (Proiectio orthogonalis). Demonstratvr in: Distantiae, angvli et volvmina in spatio.

Ad calcvlvm. Catena desinit vbi instrvmenta elementaria desinvnt. Recognitio finalis cvrsvs eam inter septem proposita generalia collocat; Calcvlvs III plana planis svperficiebvs cvrvis tangentibvs svbstitvet et determinans factore volvminis transformationis.