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Vectores in plano et aeqvationes parametricae rectae

Svmmarivm

Haec lectio vectores plani, qvi translationes repraesentant, vt paria ordinata nvmerorvm realivm introdvcit, eorvmqve svmmam, prodvctvm per nvmervm, normam et prodvctvm scalare cvm svis proprietatibvs definit. Ex prodvcto scalari dedvcvntvr condicio perpendicvlaritatis, inaeqvalitas Cavchy-Schwarziana, inaeqvalitas triangvlaris, proiectio orthogonalis et divisio angvlorvm in acvtos, rectos et obtvsos. Vectoribvs recta forma vectoriali et parametrica scribitvr, eivs vector normalis et formvla distantiae pvncti a recta invenivntvr, rectae per parametra datae secantvr, segmenta in data ratione dividvntvr et motvs aeqvabilis describitvr; problemata haec instrvmenta ad demonstrationes geometricas, ad reflexionem in specvlo et ad appropinqvationem dvorvm corporvm motorvm adhibent.

Proposita discendi

  1. Cvm vectoribvs plani operari, per componentes et geometrice, et vectorem vt combinationem linearem dvorvm aliorvm exprimere.
  2. Normas et prodvcta scalaria compvtare, eisqve vti ad vectores perpendicvlares agnoscendos, angvlos dividendos et vectorem in alivm proiciendvm.
  3. Rectam forma vectoriali, parametrica et generali scribere, ab vna forma ad aliam transire, et vectoribvs intersectiones, distantias et pvncta qvae segmentvm dividvnt compvtare.
  4. Proprietates geometricas per vectores demonstrare et motvm aeqvabilem rectis parametricis exprimere.

Vectores in plano

In lectione de aeqvatione rectae vnvmqvodqve pvnctvm plani cvm pari svarvm coordinatarvm identificatvm est. Mvltae tamen magnitvdines non pvncto sed translatione describvntvr: tribvs vnitatibvs dextrorsvm et dvabvs svrsvm progredi idem mandatvm est, qvodcvmqve pvnctvm initivm sit. Idem accidit velocitati et vi, qvae magnitvdinem, directionem et sensvm habent, et qvae in cvrsv Fvndamenta Mechanicae tractantvr. Translatio vt sagitta pingitvr, et dvobvs nvmeris determinatvr: progressv horizontali et progressv verticali. Qvae si ita svnt, definitio natvralis haec est.

Vector plani est par ordinatvm nvmerorvm realivm, u⃗=(u1,u2)\vec{u} = \left( u_1, u_2 \right); nvmeri u1u_1 et u2u_2 svnt eivs componentes. Vector qvi a pvncto P=(x1,y1)P = (x_1, y_1) ad pvnctvm Q=(x2,y2)Q = (x_2, y_2) tendit est par differentiarvm coordinatarvm eorvm, et scribitvr PQ⃗\vec{PQ}; sagitta cvivs origo est PP et extremvm QQ evm repraesentat.
PQ⃗=(x2−x1,y2−y1)\vec{PQ} = \left( x_2 - x_1, y_2 - y_1 \right)

Idem vector infinitas repraesentationes habet: dvae sagittae evndem vectorem repraesentant cvm evndem progressvm horizontalem et evndem progressvm verticalem efficivnt, id est, cvm altera ex altera translatione obtinetvr. Inter omnes vna sola ab origine OO proficiscitvr, sagitta qvae in pvncto (u1,u2)(u_1, u_2) desinit. Inde efficitvr vt OP⃗\vec{OP}, qvi vector positionis pvncti PP appellatvr, componentes habeat coordinatas pvncti PP. Posthac P+u⃗P + \vec{u} scribitvr pro pvncto ad qvod a PP translatione u⃗\vec{u} pervenitvr, id est, pro extremo sagittae qvae u⃗\vec{u} repraesentat cvm origine PP. Vector nvllvs 0⃗=(0,0)\vec{0} = \left( 0, 0 \right) respondet non moveri, et oppositvs vectoris u⃗\vec{u} est −u⃗=(−u1,−u2)-\vec{u} = \left( -u_1, -u_2 \right), translatio contraria; ita QP⃗=−PQ⃗\vec{QP} = -\vec{PQ}.

LemmaVectores aeqvales et parallelogramma

Sint PP, QQ, RR et SS pvncta plani. Aeqvalitates PQ⃗=RS⃗\vec{PQ} = \vec{RS} et PR⃗=QS⃗\vec{PR} = \vec{QS} aeqvivalent, et ambae aeqvivalent eo qvod segmenta PSPS et QRQR idem pvnctvm medivm habent. Si qvattvor pvncta in eadem recta non iacent, PQ⃗=RS⃗\vec{PQ} = \vec{RS} si et solvm si qvadrilatervm cvivs vertices svnt PP, QQ, SS et RR, hoc ordine, parallelogrammvm est.

Tres aeqvivalentiae eo redevnt vt aeqvalitas inter nvmeros coordinatim rvrsvs disponatvr; vltima theoremate de intersectione dvarvm rectarvm nititvr, qvod in lectione de recta exstat. Abscissas qvattvor pvnctorvm p1p_1, q1q_1, r1r_1 et s1s_1 appellamvs.

Demonstratio

  1. q1−p1=s1−r1↔r1−p1=s1−q1\dato{a}{q_1 - p_1 = s_1 - r_1} \leftrightarrow \resaltar{r_1 - p_1 = s_1 - q_1}

    Aeqvalitas primarvm componentivm vectorvm PQ⃗\vec{PQ} et RS⃗\vec{RS} aeqvivalet, si r1−q1r_1 - q_1 vtriqve membro additvr, aeqvalitati primarvm componentivm vectorvm PR⃗\vec{PR} et QS⃗\vec{QS}. In figvra, P=(1,0)P = (1, 0), Q=(5,1)Q = (5, 1), R=(2,3)R = (2, 3) et S=(6,4)S = (6, 4).

  2. q1−p1=s1−r1↔p1+s12=q1+r12q_1 - p_1 = s_1 - r_1 \leftrightarrow \resaltar{\frac{p_1 + s_1}{2} = \frac{q_1 + r_1}{2}}

    Eadem aeqvalitas aeqvivalet, si p1p_1 et r1r_1 vtriqve membro addvntvr et per 22 dividitvr, eo qvod pvncta media segmentorvm PSPS et QRQR eandem abscissam habent. De ordinatis eodem modo ratiocinamvr; itaqve tres envntiationes aeqvivalent.

  3. q2−p2q1−p1=s2−r2s1−r1\dato{p}{\frac{q_2 - p_2}{q_1 - p_1} = \frac{s_2 - r_2}{s_1 - r_1}}

    Ponamvs nvnc qvattvor pvncta in eadem recta non iacere et PQ⃗=RS⃗\vec{PQ} = \vec{RS}. Rectae PQPQ et RSRS eandem directionem habent: eandem declivitatem, qvotientem componentivm illivs vectoris, vel ambae verticales svnt si prima eivs componens nvlla est. Distinctae svnt, qvia, si congrverent, qvattvor pvncta in eadem recta iacerent; itaqve parallelae svnt: si verticales non svnt, per theorema de intersectione dvarvm rectarvm non verticalivm, et si svnt, qvia dvae rectae verticales distinctae se non secant.

  4. PR⃗=QS⃗→r2−p2r1−p1=s2−q2s1−q1\vec{PR} = \vec{QS} \rightarrow \resaltar{\frac{r_2 - p_2}{r_1 - p_1} = \frac{s_2 - q_2}{s_1 - q_1}}

    Per primam aeqvivalentiam etiam PR⃗=QS⃗\vec{PR} = \vec{QS}, et idem argvmentvm ostendit latera PRPR et QSQS parallela esse. Latera opposita qvadrilateri PQSRPQSR bina parallela svnt: parallelogrammvm est.

  5. RS1⃗=PQ⃗,QS1⃗=PR⃗\vec{RS_1} = \vec{PQ}, \quad \vec{QS_1} = \vec{PR}

    Contra, sit PQSRPQSR parallelogrammvm et sit S1=R+PQ⃗S_1 = R + \vec{PQ}. Per ea qvae iam demonstrata svnt, ad pvncta PP, QQ, RR et S1S_1 adhibita, pvnctvm S1S_1 in parallela ad PQPQ per RR dvcta iacet et in parallela ad PRPR per QQ dvcta.

  6. S1=S→RS⃗=PQ⃗S_1 = S \rightarrow \resaltar{\vec{RS} = \vec{PQ}}

    Hae dvae parallelae inter se parallelae non svnt, qvia PP, QQ et RR in eadem recta non iacent, et in vno solo pvncto se secant. Cvm vertex SS parallelogrammi etiam in ambabvs iaceat, cvm S1S_1 congrvit.

ExemplvmVector qvi ab vno pvncto ad altervm tendit

Compvtare vectorem qvi a P=(1,1)P = (1, 1) ad Q=(4,3)Q = (4, 3) tendit, evm ad pvnctvm R=(−2,2)R = (-2, 2) et ad originem transferre, et cvm vectore qvi a QQ ad PP tendit comparare.

Demonstratio

  1. PQ⃗=(4−1,3−1)=(3,2)\vec{PQ} = \left( \resaltar{4 - 1}, \resaltar{3 - 1} \right) = \dato{v}{\left( 3, 2 \right)}

    Coordinatas originis sagittae a coordinatis eivs extremi svbtrahimvs: translatio est 33 vnitatvm dextrorsvm et 22 svrsvm.

  2. S=R+PQ⃗=(−2+3,2+2)=(1,4)S = R + \vec{PQ} = \left( -2 + \resaltar{3}, 2 + \resaltar{2} \right) = \left( 1, 4 \right)

    Cvm origine RR, eadem translatio ad pvnctvm SS dvcit.

  3. PR⃗=(−2−1,2−1)=(−3,1)=QS⃗\vec{PR} = \left( \resaltar{-2 - 1}, \resaltar{2 - 1} \right) = \left( -3, 1 \right) = \vec{QS}

    Per lemma etiam PR⃗\vec{PR} et QS⃗\vec{QS} aeqvales svnt, et pvncta PP, QQ, SS et RR vertices parallelogrammi svnt.

  4. OA⃗=(3,2),A=(3,2)\vec{OA} = \resaltar{\left( 3, 2 \right)}, \quad A = \left( 3, 2 \right)

    Cvm origine OO, sagitta in pvncto AA desinit, cvivs coordinatae svnt componentes vectoris: PQ⃗\vec{PQ} est vector positionis pvncti AA.

  5. QP⃗=(1−4,1−3)=(−3,−2)=−PQ⃗\vec{QP} = \left( \resaltar{1 - 4}, \resaltar{1 - 3} \right) = \left( -3, -2 \right) = -\vec{PQ}

    Sagitta qvae a QQ ad PP tendit idem segmentvm sensv contrario percvrrit: vectorem oppositvm repraesentat.

Svmma vectorvm et prodvctvm per nvmervm

Svmma vectorvm u⃗=(u1,u2)\vec{u} = \left( u_1, u_2 \right) et v⃗=(v1,v2)\vec{v} = \left( v_1, v_2 \right) est u⃗+v⃗=(u1+v1,u2+v2)\vec{u} + \vec{v} = \left( u_1 + v_1, u_2 + v_2 \right), et prodvctvm nvmeri realis tt per u⃗\vec{u} est tu⃗=(tu1,tu2)t \vec{u} = \left( t u_1, t u_2 \right). Differentia vectorvm u⃗\vec{u} et v⃗\vec{v} est u⃗−v⃗=u⃗+(−v⃗)\vec{u} - \vec{v} = \vec{u} + \left( -\vec{v} \right).

Svmma respondet translationibvs alteri post alteram peractis: si u⃗\vec{u} percvrritvr et deinde v⃗\vec{v}, progressvs horizontalis totvs est u1+v1u_1 + v_1 et verticalis u2+v2u_2 + v_2. Qvod ad pvncta attinet, pro tribvs qvibvslibet pvnctis AA, BB et CC valet relatio Chaslesiana, qvia differentiae coordinatarvm alia aliam excipivnt: (b1−a1)+(c1−b1)=c1−a1(b_1 - a_1) + (c_1 - b_1) = c_1 - a_1, et similiter de ordinatis.

AB⃗+BC⃗=AC⃗\vec{AB} + \vec{BC} = \vec{AC}

Inde dvae constrvctiones svmmae orivntvr: regvla triangvli, qvae originem vectoris v⃗\vec{v} in extremo vectoris u⃗\vec{u} collocat, et regvla parallelogrammi, qvae ambos vectores ab eadem origine pingit et diagonalem parallelogrammi qvod determinant svmit. Praesertim PQ⃗=OQ⃗−OP⃗\vec{PQ} = \vec{OQ} - \vec{OP}: omnis vector inter dvo pvncta est differentia vectorvm positionis eorvm. Prodvctvm tu⃗t \vec{u} vero ambos progressvs per tt mvltiplicat: si t>0t > 0, directionem et sensvm vectoris u⃗\vec{u} servat; si t<0t < 0, directionem servat et sensvm invertit; et, vt in norma videbitvr, longitvdinem per ∣t∣\abs{t} mvltiplicat.

ExemplvmSvmma, differentia et regvla parallelogrammi

Sint u⃗=(4,1)\vec{u} = \left( 4, 1 \right) et v⃗=(1,3)\vec{v} = \left( 1, 3 \right). Constrvere u⃗+v⃗\vec{u} + \vec{v} regvla triangvli et regvla parallelogrammi, et constrvere u⃗−v⃗\vec{u} - \vec{v}.

Demonstratio

  1. u⃗=(4,1),v⃗=(1,3)\dato{u}{\vec{u} = \left( 4, 1 \right)}, \quad \dato{v}{\vec{v} = \left( 1, 3 \right)}

    Ambos vectores ab origine OO pingimvs.

  2. u⃗+v⃗=(4+1,1+3)=(5,4)\vec{u} + \vec{v} = \left( \resaltar{4 + 1}, \resaltar{1 + 3} \right) = \dato{s}{\left( 5, 4 \right)}

    Regvla triangvli: cvm origine in extremo vectoris u⃗\vec{u}, v⃗\vec{v} ad pvnctvm (5,4)(5, 4) dvcit; svmma per singvlas componentes compvtatvr.

  3. v⃗+u⃗=(5,4)=u⃗+v⃗\vec{v} + \vec{u} = \resaltar{\left( 5, 4 \right)} = \vec{u} + \vec{v}

    Regvla parallelogrammi: si primvm v⃗\vec{v} percvrritvr et deinde u⃗\vec{u}, ad idem pvnctvm pervenitvr. Svmma est diagonalis parallelogrammi cvivs latera svnt u⃗\vec{u} et v⃗\vec{v}, qvae a OO proficiscitvr.

  4. −v⃗=(−1,−3)=(−1)v⃗-\vec{v} = \left( \resaltar{-1}, \resaltar{-3} \right) = \left( -1 \right) \vec{v}

    Oppositvs vectoris v⃗\vec{v} est eivs prodvctvm per −1-1: eandem longitvdinem et sensvm contrarivm habet.

  5. u⃗−v⃗=(4−1,1−3)=(3,−2)\vec{u} - \vec{v} = \left( \resaltar{4 - 1}, \resaltar{1 - 3} \right) = \left( 3, -2 \right)

    Differentia est svmma vectorvm u⃗\vec{u} et −v⃗-\vec{v}. Translata, est sagitta qvae ab extremo vectoris v⃗\vec{v} ad extremvm vectoris u⃗\vec{u} tendit: altera diagonalis parallelogrammi.

Dvae operationes a nvmeris realibvs proprietates infra envmeratas hereditate accipivnt, qvae valent pro omnibvs vectoribvs u⃗\vec{u}, v⃗\vec{v} et w⃗\vec{w} et omnibvs nvmeris realibvs ss et tt:

  1. Commvtativitas svmmae: u⃗+v⃗=v⃗+u⃗\vec{u} + \vec{v} = \vec{v} + \vec{u}.
  2. Associativitas svmmae: (u⃗+v⃗)+w⃗=u⃗+(v⃗+w⃗)\left( \vec{u} + \vec{v} \right) + \vec{w} = \vec{u} + \left( \vec{v} + \vec{w} \right).
  3. Elementvm nevtrvm: u⃗+0⃗=u⃗\vec{u} + \vec{0} = \vec{u}.
  4. Elementvm oppositvm: u⃗+(−u⃗)=0⃗\vec{u} + \left( -\vec{u} \right) = \vec{0}.
  5. Associativitas prodvcti: s(tu⃗)=(st)u⃗s \left( t \vec{u} \right) = \left( s t \right) \vec{u}.
  6. Nvmervs vnvs nevtrvm: 1u⃗=u⃗1 \vec{u} = \vec{u}.
  7. Distribvtivitas respectv svmmae vectorvm: t(u⃗+v⃗)=tu⃗+tv⃗t \left( \vec{u} + \vec{v} \right) = t \vec{u} + t \vec{v}.
  8. Distribvtivitas respectv svmmae nvmerorvm: (s+t)u⃗=su⃗+tu⃗\left( s + t \right) \vec{u} = s \vec{u} + t \vec{u}.
TheoremaProprietates operationvm cvm vectoribvs

Pro omnibvs vectoribvs u⃗\vec{u}, v⃗\vec{v} et w⃗\vec{w} plani et omnibvs nvmeris realibvs ss et tt valent octo proprietates indicis svperioris.

Dvae operationes per singvlas componentes agvnt, ita vt qvaeqve proprietas ad proprietatem respondentem nvmerorvm realivm redeat, in singvlis componentibvs adhibitam. Satis est primam componentem scribere, qvia secvnda eadem est cvm indice 22.

Demonstratio

  1. u1+v1=v1+u1,(u1+v1)+w1=u1+(v1+w1)\resaltar{u_1 + v_1} = \resaltar{v_1 + u_1}, \qquad \resaltar{\left( u_1 + v_1 \right) + w_1} = \resaltar{u_1 + \left( v_1 + w_1 \right)}

    Commvtativitas et associativitas: ab eis svmmae nvmerorvm realivm hereditate accipivntvr.

  2. u1+0=u1,u1+(−u1)=0\resaltar{u_1 + 0} = \resaltar{u_1}, \qquad \resaltar{u_1 + \left( -u_1 \right)} = \resaltar{0}

    Nevtrvm et oppositvm: 00 est nevtrvm svmmae et −u1-u_1 est oppositvs vectoris u1u_1.

  3. s(tu1)=(st)u1,1⋅u1=u1\resaltar{s \left( t u_1 \right)} = \resaltar{\left( s t \right) u_1}, \qquad \resaltar{1 \cdot u_1} = \resaltar{u_1}

    Associativitas prodvcti et nvmervs vnvs: prodvctvm nvmerorvm realivm associativvm est et 11 pro nevtro habet.

  4. t(u1+v1)=tu1+tv1,(s+t)u1=su1+tu1\resaltar{t \left( u_1 + v_1 \right)} = \resaltar{t u_1 + t v_1}, \qquad \resaltar{\left( s + t \right) u_1} = \resaltar{s u_1 + t u_1}

    Dvae distribvtivitates: prodvctvm nvmerorvm realivm respectv svmmae distribvtivvm est.

  5. u⃗+v⃗=(u1+v1,u2+v2)=(v1+u1,v2+u2)=v⃗+u⃗\vec{u} + \vec{v} = \left( \resaltar{u_1 + v_1}, \resaltar{u_2 + v_2} \right) = \left( \resaltar{v_1 + u_1}, \resaltar{v_2 + u_2} \right) = \vec{v} + \vec{u}

    Dvo vectores aeqvales svnt si et solvm si eorvm dvae componentes aeqvales svnt; itaqve octo aeqvalitates inter vectores valent. Exempli gratia, prima.

Hae octo proprietates svnt axiomata spatii vectorialis: omnis copia cvivs elementa inter se addvntvr et per nvmeros reales mvltiplicantvr, octo his regvlis servatis, spativm vectoriale appellatvr, et qvidqvid ex eis sine componentibvs dedvcitvr in qvolibet eorvm valebit. Polynomia lectionis de algebra polynomiorvm, cvm sva svmma et svo prodvcto per nvmervm, alivd spativm vectoriale efficivnt.

Combinationes lineares et vectores paralleli

Combinatio linearis vectorvm u⃗\vec{u} et v⃗\vec{v} est vector formae su⃗+tv⃗s \vec{u} + t \vec{v}, vbi ss et tt reales svnt. Vectores e⃗1=(1,0)\vec{e}_1 = \left( 1, 0 \right) et e⃗2=(0,1)\vec{e}_2 = \left( 0, 1 \right), qvi vna vnitate in directione cvivsqve axis progredivntvr, permittvnt omnem vectorem vt eorvm combinationem scribere, et vno tantvm modo, qvia ae⃗1+be⃗2=(a,b)a \vec{e}_1 + b \vec{e}_2 = \left( a, b \right) aeqvalis est u⃗\vec{u} solvm si a=u1a = u_1 et b=u2b = u_2. Ideo basis canonica plani appellantvr:

u⃗=(u1,u2)=u1e⃗1+u2e⃗2\vec{u} = \left( u_1, u_2 \right) = u_1 \vec{e}_1 + u_2 \vec{e}_2

Dvo vectores non nvlli u⃗\vec{u} et v⃗\vec{v} paralleli svnt cvm alter mvltiplex alterivs est, id est, cvm v⃗=tu⃗\vec{v} = t \vec{u} pro aliqvo nvmero reali tt, qvi nvllvs esse non potest; evndem sensvm habent si t>0t > 0 et sensvs contrarios si t<0t < 0. Sagittae qvae eos ab eadem origine repraesentant tvm in vna eademqve recta iacent.

LemmaCriterivm parallelismi

Dvo vectores non nvlli u⃗\vec{u} et v⃗\vec{v} paralleli svnt si et solvm si u1v2−u2v1=0u_1 v_2 - u_2 v_1 = 0.

Altera implicatio compvtatio est; ad alteram nvmervs tt cvm v⃗=tu⃗\vec{v} = t \vec{u} fabricandvs est, per componentem non nvllam vectoris u⃗\vec{u} dividendo.

Demonstratio

  1. u1(tu2)−u2(tu1)=0u_1 \resaltar{\left( t u_2 \right)} - u_2 \resaltar{\left( t u_1 \right)} = 0

    Si v⃗=tu⃗\vec{v} = t \vec{u}, componentes vectoris v⃗\vec{v} svnt tu1t u_1 et tu2t u_2, et expressio evanescit. In figvra, u⃗=(2,1)\vec{u} = \left( 2, 1 \right) et v⃗=(6,3)=3u⃗\vec{v} = \left( 6, 3 \right) = 3 \vec{u}.

  2. u1v2=u2v1,t=v1u1→v1=tu1\dato{h}{u_1 v_2 = u_2 v_1}, \quad t = \frac{v_1}{u_1} \rightarrow \resaltar{v_1 = t u_1}

    Contra, ponamvs expressionem evanescere, et primvm ponamvs u1≠0u_1 \neq 0. Definimvs tt vt qvotientem primarvm componentivm, ita vt prima componens vectoris v⃗\vec{v} sit tu1t u_1. In figvra, t=62=3t = \frac{6}{2} = 3.

  3. v2=u2v1u1=u2tu1u1=tu2v_2 = \frac{u_2 \resaltar{v_1}}{u_1} = \frac{u_2 \resaltar{t u_1}}{u_1} = t u_2

    Ex hypothesi v2v_2 explicamvs: etiam secvnda componens est tu2t u_2, ideoqve v⃗=tu⃗\vec{v} = t \vec{u}. In figvra, 3=3⋅13 = 3 \cdot 1.

  4. u1=0→v1=0,t=v2u2u_1 = 0 \rightarrow \resaltar{v_1 = 0}, \quad t = \frac{v_2}{u_2}

    Si u1=0u_1 = 0, tvm u2≠0u_2 \neq 0, qvia u⃗\vec{u} nvllvs non est; hypothesis dat u2v1=0u_2 v_1 = 0, vnde v1=0v_1 = 0, et qvotiente secvndarvm componentivm efficitvr v⃗=tu⃗\vec{v} = t \vec{u}. In figvra, u⃗=(0,2)\vec{u} = \left( 0, 2 \right) et v⃗=(0,−3)\vec{v} = \left( 0, -3 \right), cvm t=−32t = -\frac{3}{2}.

Nvmervs u1v2−u2v1u_1 v_2 - u_2 v_1 est determinans lectionis de systematibvs linearibvs, et lemma infra positvm cavsam explicat.

LemmaCombinationes dvorvm vectorvm non parallelorvm

Si u⃗\vec{u} et v⃗\vec{v} vectores non nvlli et non paralleli svnt, omnis w⃗\vec{w} combinatio linearis eorvm est vno tantvm modo: exstant nvmeri ss et tt, et vnvm tantvm par, tales vt w⃗=su⃗+tv⃗\vec{w} = s \vec{u} + t \vec{v}.

Aeqvalitas qvaesita, per componentes scripta, est systema dvarvm aeqvationvm linearivm cvm incognitis ss et tt, et satis est ei regvlam Crameri adhibere.

Demonstratio

  1. {u1s+v1t=w1u2s+v2t=w2\begin{cases} u_1 s + v_1 t = w_1 \\ u_2 s + v_2 t = w_2 \end{cases}

    Aeqvalitatem su⃗+tv⃗=w⃗s \vec{u} + t \vec{v} = \vec{w} per singvlas componentes scribimvs. In figvra, u⃗=(2,1)\vec{u} = \left( 2, 1 \right), v⃗=(1,3)\vec{v} = \left( 1, 3 \right) et w⃗=(4,7)\vec{w} = \left( 4, 7 \right).

  2. D=u1v2−u2v1≠0\dato{D}{D = u_1 v_2 - u_2 v_1 \neq 0}

    Determinans systematis nvllvm non est, per criterivm parallelismi, qvia vectores paralleli non svnt. In figvra, 2⋅3−1⋅1=52 \cdot 3 - 1 \cdot 1 = 5.

  3. s=w1v2−v1w2D,t=u1w2−w1u2Ds = \frac{w_1 v_2 - v_1 w_2}{\resaltar{D}}, \quad t = \frac{u_1 w_2 - w_1 u_2}{\resaltar{D}}

    Per regvlam Crameri systema vnam omnino solvtionem habet, qvotientibvs infra positis datam. In figvra, s=12−75=1s = \frac{12 - 7}{5} = 1 et t=14−45=2t = \frac{14 - 4}{5} = 2, et revera w⃗=u⃗+2v⃗\vec{w} = \vec{u} + 2 \vec{v}.

  4. su⃗+tv⃗=s1u⃗+t1v⃗→s=s1∧t=t1s \vec{u} + t \vec{v} = s_1 \vec{u} + t_1 \vec{v} \rightarrow \resaltar{s = s_1 \wedge t = t_1}

    Praesertim, si dvae combinationes vectorvm u⃗\vec{u} et v⃗\vec{v} aeqvales svnt, ambae svnt solvtio systematis cvm eodem w⃗\vec{w}, et earvm coefficientes congrvvnt. Haec comparatio coefficientivm instrvmentvm est plvrivm demonstrationvm hvivs lectionis.