Ad argvmentvm transi
Topos Uranos
Latine

Distantiae, angvli et volvmina in spatio

Svmmarivm

Haec lectio instrvmenta trivm lectionvm proxime svperiorvm, prodvctvm scalare, prodvctvm vectoriale et prodvctvm mixtvm, cvm aeqvationibvs rectarvm et planorvm conivngit, vt in spatio metiamvr. Distantiam pvncti a copia definit et, proiciendo in normalem, formvlam distantiae pvncti a plano demonstrat et, prodvcto vectoriali adhibito, formvlam distantiae pvncti a recta, vtramqve cvm pede perpendicvlaris tamqvam vnico pvncto proximo. Ex eis distantias inter plana parallela et inter rectas parallelas dedvcit, et demonstrat dvas rectas non coplanares vnicam perpendicvlarem commvnem habere, qvae systemate dvarvm aeqvationvm solvto invenitvr et cvivs longitvdo est prodvctvm mixtvm per normam divisvm. Angvlos inter dvas rectas, inter rectam et planvm et inter dvo plana definit et compvtat, et demonstrat eos cvm angvlis geometriae elementaris congrvere; planvm medivm perpendicvlare segmenti constrvit et pvncta symmetrica pvncti respectv plani et rectae; demonstrat aream proiectionis polygoni plani esse aream eivs per cosinvm angvli dihedralis mvltiplicatam, et sphaeram tractat, eivs aeqvationem, eivs intersectionem cvm plano et eivs planvm tangens. Applicationes svnt plana crystalli eorvmqve indices Milleriani et locatio per satellites; problemata tractant de tetraedro regvlari, de pvnctis a tribvs pvnctis aeqve distantibvs, de vmbra tabvlae solaris, de planis sphaerae tangentibvs qvae rectam continent et de dvabvs aeronavibvs qvarvm itinera non coplanaria svnt.

Proposita discendi

  1. Distantiam pvncti a plano et a recta, inter plana parallela, inter rectas parallelas et inter rectas non coplanares compvtare, vnamqvamqve formvlam demonstrando et pvncta inveniendo in qvibvs attingitvr.
  2. Perpendicvlarem commvnem dvarvm rectarvm non coplanarivm invenire et demonstrare eam vnicam esse.
  3. Angvlos inter rectas, inter rectam et planvm et inter dvo plana compvtare, et demonstrare eos cvm angvlis geometricis qvos definivnt congrvere.
  4. Planvm medivm perpendicvlare segmenti et pvncta symmetrica pvncti respectv plani et rectae constrvere.
  5. Areas et volvmina prodvctis vectoriali et mixto compvtare, et relationem inter aream polygoni plani et aream proiectionis eivs demonstrare et adhibere.
  6. Centrvm et radivm sphaerae ex eivs aeqvatione ervere, positionem eivs respectv plani diivdicare, circvmferentiam sectionis et planvm tangens invenire, et his instrvmentis problemata crystallographiae, energiae solaris, navigationis aeriae et locationis per satellites describere.

Distantia pvncti a plano et a recta

Lectio de vectoribvs in plano distantiam pvncti a recta plani obtinvit proiciendo in vectorem normalem. In spatio, lectio de rectis et planis in spatio planvm per P0P_0 transiens cvm vectore normali n⃗=(a,b,c)\vec{n} = \left( a, b, c \right) non nvllo scripsit vt copiam pvnctorvm XX cvm n⃗⋅P0X⃗=0\vec{n} \cdot \vec{P_0 X} = 0, id est cvm aeqvatione ax+by+cz=dax + by + cz = d, et rectam per AA transevntem cvm vectore directore u⃗\vec{u} vt copiam pvnctorvm A+tu⃗A + t \vec{u}, cvm tt reali. Si hoc ita est, notio plani ad plana spatii sine mvtatione valet; pro rectis avtem normalis non iam vna directio est, et prodvctvm vectoriale lectionis de prodvcto vectoriali et prodvcto mixto eivs locvm tenet. Facta geometriae Evclideae spatii qvae adhibentvr svnt ea qvae in lectione de vectoribvs in spatio et prodvcto scalari admissa svnt.

Sit FF copia non vacva pvnctorvm spatii et PP pvnctvm. Si inter distantias ipsivs PP a pvnctis ipsivs FF est vna qvae minor est vel aeqvalis omnibvs ceteris, is nvmervs est distantia ipsivs PP a FF, et scribitvr d⁡(P,F)\operatorname{d}(P, F). Eodem modo, si inter distantias pvnctorvm copiae FF a pvnctis alterivs copiae GG est vna minor vel aeqvalis omnibvs, is nvmervs est distantia inter FF et GG, d⁡(F,G)\operatorname{d}(F, G).

Definitio non praestat distantiam exsistere: inter infinitas distantias fieri potest vt nvlla minima sit. Ideo theoremata qvae infra dantvr non solvm formvlam praebent; demonstrant pvnctvm proximvm esse, idqve vnicvm.

TheoremaDistantia pvncti a plano

Sit π\pi planvm ax+by+cz=dax + by + cz = d, cvm n⃗=(a,b,c)\vec{n} = \left( a, b, c \right) non nvllo, et sit P=(x0,y0,z0)P = (x_0, y_0, z_0). Pvnctvm H=P−λn⃗H = P - \lambda \vec{n}, cvm λ=ax0+by0+cz0−d∥n⃗∥2\lambda = \frac{a x_0 + b y_0 + c z_0 - d}{{\norma{\vec{n}}}^2}, in π\pi iacet, HP⃗\vec{HP} ad π\pi normalis est, et HH est vnicvm pvnctvm ipsivs π\pi ipsi PP proximvm. Itaqve:

d⁡(P,π)=∣ax0+by0+cz0−d∣a2+b2+c2\operatorname{d}(P, \pi) = \frac{\abs{a x_0 + b y_0 + c z_0 - d}}{\sqrt{a^2 + b^2 + c^2}}

Vector qvi a qvovis pvncto plani ad PP dvcit in normalem proicitvr, vt in plano; minimalitas est theorema Pythagoricvm. Figvra adhibet planvm 2x+y+2z=42x + y + 2z = 4 et pvnctvm P=(2,3,3)P = (2, 3, 3).

Demonstratio

  1. p⃗=QP⃗⋅n⃗∥n⃗∥2n⃗\vec{p} = \dato{p}{\frac{\vec{QP} \cdot \vec{n}}{{\norma{\vec{n}}}^2} \vec{n}}

    Eligimvs pvnctvm Q=(q1,q2,q3)Q = (q_1, q_2, q_3) ipsivs π\pi et QP⃗\vec{QP} in n⃗\vec{n} proicimvs theoremate proiectionis orthogonalis lectionis de vectoribvs in spatio. In figvra, Q=(2,0,0)Q = (2, 0, 0).

  2. QP⃗⋅n⃗=ax0+by0+cz0−(aq1+bq2+cq3)=ax0+by0+cz0−d\vec{QP} \cdot \vec{n} = a x_0 + b y_0 + c z_0 \resaltar{- \left( a q_1 + b q_2 + c q_3 \right)} = \dato{e}{a x_0 + b y_0 + c z_0 - \resaltar{d}}

    Cvm QQ in π\pi iaceat, aq1+bq2+cq3=da q_1 + b q_2 + c q_3 = d, et prodvctvm scalare nvmeratoris est primvm membrvm aeqvationis in PP aestimatvm, minvs dd.

  3. QH⃗=QP⃗−λn⃗,n⃗⋅QH⃗=0→H∈π\vec{QH} = \vec{QP} - \resaltar{\lambda \vec{n}}, \qquad \vec{n} \cdot \vec{QH} = 0 \rightarrow \dato{h}{H \in \pi}

    Ergo p⃗=λn⃗\vec{p} = \lambda \vec{n} et H=P−p⃗H = P - \vec{p}. Vector QH⃗=QP⃗−p⃗\vec{QH} = \vec{QP} - \vec{p} est componens orthogonalis proiectionis, ad n⃗\vec{n} perpendicvlaris; per aeqvationem normalem plani, HH in π\pi iacet.

  4. ∥PX⃗∥2=∥PH⃗∥2+∥HX⃗∥2≥∥PH⃗∥2{\norma{\vec{PX}}}^2 = {\norma{\vec{PH}}}^2 + \resaltar{{\norma{\vec{HX}}}^2} \ge {\norma{\vec{PH}}}^2

    Sit XX qvodvis pvnctvm ipsivs π\pi. Per relationem Chaslesianam, PX⃗=PH⃗+HX⃗\vec{PX} = \vec{PH} + \vec{HX}; primvs addendvs mvltiplex est ipsivs n⃗\vec{n} et secvndvs dvo pvncta ipsivs π\pi ivngit, ita vt orthogonales sint, et aeqvalitas Pythagorica valet.

  5. d⁡(P,π)=∥p⃗∥=∣QP⃗⋅n⃗∣∥n⃗∥=∣ax0+by0+cz0−d∣a2+b2+c2\operatorname{d}(P, \pi) = \norma{\vec{p}} = \frac{\resaltar{\abs{\vec{QP} \cdot \vec{n}}}}{\norma{\vec{n}}} = \frac{\abs{a x_0 + b y_0 + c z_0 - d}}{\sqrt{a^2 + b^2 + c^2}}

    Aeqvalitas tantvm obtinet si HX⃗=0⃗\vec{HX} = \vec{0}, id est si X=HX = H: pes perpendicvlaris est vnicvm pvnctvm proximvm, et distantia est norma proiectionis.

  6. d⁡(P,π)=∣2⋅2+3+2⋅3−4∣4+1+4=93=3\operatorname{d}(P, \pi) = \frac{\abs{2 \cdot 2 + 3 + 2 \cdot 3 - 4}}{\sqrt{4 + 1 + 4}} = \frac{\resaltar{9}}{\resaltar{3}} = 3

    In figvra, QP⃗=(0,3,3)\vec{QP} = \left( 0, 3, 3 \right), QP⃗⋅n⃗=9\vec{QP} \cdot \vec{n} = 9 et ∥n⃗∥=3\norma{\vec{n}} = 3, ita vt λ=1\lambda = 1, pes sit H=(0,2,1)H = (0, 2, 1) et distantia 33.

Signvm qvoqve nvmeri ax0+by0+cz0−da x_0 + b y_0 + c z_0 - d, qvod formvla valore absolvto occvltat, significationem habet. Qvoniam P=H+λn⃗P = H + \lambda \vec{n}, positivvm est cvm ad PP a plano pervenitvr movendo in sensv ipsivs n⃗\vec{n}, et negativvm cvm in sensv contrario pervenitvr. Pvncta cvm ax+by+cz>dax + by + cz > d vnvm e dvobvs semispatiis qvae planvm determinat efficivnt, et pvncta cvm ax+by+cz<dax + by + cz < d altervm; dvo pvncta in eodem latere plani svnt si et solvm si expressio in vtroqve idem signvm habet. Sic origo et pvnctvm (2,3,3)(2, 3, 3) in lateribvs oppositis plani 2x+y+2z=42x + y + 2z = 4 svnt, qvia 0−4<0<90 - 4 < 0 < 9.

TheoremaDistantia pvncti a recta

Sit rr recta A+tu⃗A + t \vec{u}, cvm u⃗\vec{u} non nvllo, et sit PP pvnctvm. Pvnctvm H=A+t0u⃗H = A + t_0 \vec{u}, cvm t0=AP⃗⋅u⃗∥u⃗∥2t_0 = \frac{\vec{AP} \cdot \vec{u}}{{\norma{\vec{u}}}^2}, est vnicvm pvnctvm ipsivs rr ipsi PP proximvm, HP⃗\vec{HP} ad u⃗\vec{u} perpendicvlaris est, et:

d⁡(P,r)=∥AP⃗×u⃗∥∥u⃗∥\operatorname{d}(P, r) = \frac{\norma{\vec{AP} \times \vec{u}}}{\norma{\vec{u}}}

Vector AP⃗\vec{AP} in proiectionem in u⃗\vec{u} et in componentem orthogonalem dividitvr; norma hvivs identitate Lagrangiana exprimitvr. Figvra adhibet rectam per A=(0,0,1)A = (0, 0, 1) transevntem cvm vectore directore u⃗=(1,2,2)\vec{u} = \left( 1, 2, 2 \right) et pvnctvm P=(3,3,1)P = (3, 3, 1).

Demonstratio

  1. AP⃗=t0u⃗+w⃗,w⃗⋅u⃗=0\vec{AP} = t_0 \vec{u} + \dato{w}{\vec{w}}, \qquad \vec{w} \cdot \vec{u} = 0

    Per theorema proiectionis orthogonalis, AP⃗\vec{AP} est svmma proiectionis svae t0u⃗t_0 \vec{u} in u⃗\vec{u} et vectoris w⃗\vec{w} ipsi u⃗\vec{u} orthogonalis. Pvnctvm H=A+t0u⃗H = A + t_0 \vec{u} in rr iacet et HP⃗=w⃗\vec{HP} = \vec{w}.

  2. XP⃗=w⃗+(t0−t)u⃗,∥XP⃗∥2=∥w⃗∥2+(t0−t)2∥u⃗∥2≥∥w⃗∥2\vec{XP} = \vec{w} + \resaltar{\left( t_0 - t \right) \vec{u}}, \qquad {\norma{\vec{XP}}}^2 = {\norma{\vec{w}}}^2 + \resaltar{\left( t_0 - t \right)^2 {\norma{\vec{u}}}^2} \ge {\norma{\vec{w}}}^2

    Pro qvovis pvncto X=A+tu⃗X = A + t \vec{u} ipsivs rr, vector XP⃗\vec{XP} est w⃗\vec{w} plvs mvltiplex ipsivs u⃗\vec{u}; dvo addendi orthogonales svnt et aeqvalitas Pythagorica valet.

  3. d⁡(P,r)2=∥w⃗∥2=∥AP⃗∥2−(AP⃗⋅u⃗)2∥u⃗∥2=∥AP⃗∥2∥u⃗∥2−(AP⃗⋅u⃗)2∥u⃗∥2{\operatorname{d}(P, r)}^2 = {\norma{\vec{w}}}^2 = {\norma{\vec{AP}}}^2 - \resaltar{\frac{\left( \vec{AP} \cdot \vec{u} \right)^2}{{\norma{\vec{u}}}^2}} = \dato{q}{\frac{{\norma{\vec{AP}}}^2 {\norma{\vec{u}}}^2 - \left( \vec{AP} \cdot \vec{u} \right)^2}{{\norma{\vec{u}}}^2}}

    Aeqvalitas tantvm obtinet si t=t0t = t_0: HH est vnicvm pvnctvm proximvm et distantia est norma ipsivs w⃗\vec{w}, cvivs qvadratvm aeqvalitas Pythagorica proiectionis dat.

  4. d⁡(P,r)2=∥AP⃗×u⃗∥2∥u⃗∥2→d⁡(P,r)=∥AP⃗×u⃗∥∥u⃗∥{\operatorname{d}(P, r)}^2 = \frac{\resaltar{{\norma{\vec{AP} \times \vec{u}}}^2}}{{\norma{\vec{u}}}^2} \rightarrow \operatorname{d}(P, r) = \frac{\norma{\vec{AP} \times \vec{u}}}{\norma{\vec{u}}}

    Per identitatem Lagrangianam lectionis de prodvcto vectoriali, nvmerator est qvadratvm normae ipsivs AP⃗×u⃗\vec{AP} \times \vec{u}; radix extrahitvr.

  5. AP⃗×u⃗=(6,−6,3),d⁡(P,r)=93=3\vec{AP} \times \vec{u} = \left( 6, -6, 3 \right), \qquad \operatorname{d}(P, r) = \frac{\resaltar{9}}{\resaltar{3}} = 3

    In figvra, AP⃗=(3,3,0)\vec{AP} = \left( 3, 3, 0 \right) et AP⃗⋅u⃗=9\vec{AP} \cdot \vec{u} = 9, ita vt t0=1t_0 = 1, H=(1,2,3)H = (1, 2, 3) et w⃗=(2,1,−2)\vec{w} = \left( 2, 1, -2 \right), normae 33. Prodvctvm vectoriale idem dat.

Formvla lectionem geometricam directam habet: ∥AP⃗×u⃗∥\norma{\vec{AP} \times \vec{u}} est area parallelogrammi latervm AP⃗\vec{AP} et u⃗\vec{u}, et ∥u⃗∥\norma{\vec{u}} est longitvdo basis eivs in recta; qvotiens est altitvdo, qvae est distantia qvaesita. Cvm sola distantia qvaeritvr, prodvctvm vectoriale vitat ne pes perpendicvlaris compvtetvr.