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Prodvctvm vectoriale et prodvctvm mixtvm

Svmmarivm

Haec lectio prodvctvm vectoriale dvorvm vectorvm spatii per eorvm coordinatas definit, vt determinantem formalem cvm vectoribvs i⃗\vec{i}, j⃗\vec{j} et k⃗\vec{k}, et proprietatibvs determinantivm demonstrat id anticommvtativvm et bilineare esse, non avtem associativvm. Demonstrat prodvctvm vectoriale ad ambos factores svos perpendicvlare esse, identitatem Lagrangianam et, ex ea, normam eivs aream parallelogrammi a factoribvs determinati esse atqve id evanescere tvm demvm cvm factores paralleli svnt. Formvlam dvpli prodvcti vectorialis et identitatem Iacobianam statvit, prodvctvm mixtvm vt determinantem ordinis tertii definit, proprietatem eivs cyclicam ostendit et id vt volvmen parallelepipedi et tetraedri interpretatvr, cvm criterio complanaritatis et expressione volvminis per prodvcta scalaria. Accvrate declarat qvid in regvla manvs dextrae demonstretvr et qvid conventio sit, ostendit prodvctvm vectoriale esse partem vectorialem prodvcti dvorvm qvaternionvm pvrorvm, idqve adhibet ad momentvm vis, ad vim magneticam in onvs electricvm motvm, ad aream triangvli in spatio et ad normales graphices compvtatoriae.

Proposita discendi

  1. Prodvcta vectorialia et prodvcta mixta per coordinatas et per determinantes compvtare, et cvm eis operari adhibendo anticommvtativitatem, bilinearitatem, proprietatem cyclicam et formvlam dvpli prodvcti vectorialis.
  2. Demonstrare perpendicvlaritatem prodvcti vectorialis, identitatem Lagrangianam, criterivm parallelismi, formvlam dvpli prodvcti vectorialis et criterivm complanaritatis.
  3. Areas parallelogrammorvm et triangvlorvm in spatio et volvmina parallelepipedorvm et tetraedrorvm compvtare, et decernere vtrvm qvattvor pvncta complanaria sint necne.
  4. In regvla manvs dextrae distingvere qvid demonstretvr a qvo convenitvr, et prodvcto vectoriali momentvm vis, vim magneticam et orientationem svperficiervm in graphice compvtatoria repraesentare.

Prodvctvm vectoriale

In lectione de vectoribvs in spatio et prodvcto scalari vectores ipsivs R3\mathbb{R}^3 vt terniones u⃗=(u1,u2,u3)\vec{u} = \left( u_1, u_2, u_3 \right) scripti svnt, et prodvctvm scalare u⃗⋅v⃗=u1v1+u2v2+u3v3\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 longitvdines et angvlos metiri, de perpendicvlaritate decernere et vectorem in alivm proicere permisit. Prodvctvm scalare dvorvm vectorvm nvmervs est. Atqvi mvlta spatii problemata vectorem postvlant: directionem ad svperficiem planam perpendicvlarem, axem circa qvem vis corpvs vertit, directionem in qvam campvs magneticvs onvs electricvm deflectit. In plano qvaeqve directio vnam tantvm perpendicvlarem habet; in spatio perpendicvlares ad directionem planvm totvm implent, et dvae directiones non parallelae reqvirvntvr vt perpendicvlaris commvnis figatvr. Qvod si ita est, qvaestio natvralis haec est: datis dvobvs vectoribvs u⃗\vec{u} et v⃗\vec{v}, qvis vector x⃗=(x1,x2,x3)\vec{x} = \left( x_1, x_2, x_3 \right) condiciones u⃗⋅x⃗=0\vec{u} \cdot \vec{x} = 0 et v⃗⋅x⃗=0\vec{v} \cdot \vec{x} = 0 explet?

Dvae svnt aeqvationes lineares cvm tribvs incognitis. Si prima, u1x1+u2x2+u3x3=0u_1 x_1 + u_2 x_2 + u_3 x_3 = 0, per v1v_1 mvltiplicatvr, secvnda per u1u_1, et altera ab altera svbtrahitvr, evanescit x1x_1 et manet (u2v1−u1v2)x2+(u3v1−u1v3)x3=0\left( u_2 v_1 - u_1 v_2 \right) x_2 + \left( u_3 v_1 - u_1 v_3 \right) x_3 = 0, qvae expletvr cvm x2=u3v1−u1v3x_2 = u_3 v_1 - u_1 v_3 et x3=u1v2−u2v1x_3 = u_1 v_2 - u_2 v_1; eodem modo cvm ceteris incognitis ratiocinando apparet x1=u2v3−u3v2x_1 = u_2 v_3 - u_3 v_2. Tres expressiones svnt determinantes ordinis secvndi ex componentibvs ipsorvm u⃗\vec{u} et v⃗\vec{v} formati, sicvt ii lectionis de determinantibvs. Haec inqvisitio definitionem svggerit; capvt «Geometria prodvcti vectorialis» demonstrat vectorem sic formatvm revera ad ambos factores perpendicvlarem esse.

Sint u⃗=(u1,u2,u3)\vec{u} = \left( u_1, u_2, u_3 \right) et v⃗=(v1,v2,v3)\vec{v} = \left( v_1, v_2, v_3 \right) vectores ipsivs R3\mathbb{R}^3. Eorvm prodvctvm vectoriale, sive prodvctvm crvcis, est vector u⃗×v⃗=(u2v3−u3v2,u3v1−u1v3,u1v2−u2v1)\vec{u} \times \vec{v} = \left( u_2 v_3 - u_3 v_2, u_3 v_1 - u_1 v_3, u_1 v_2 - u_2 v_1 \right).

Definitio regvlaritatem habet qvam retinere expedit. Qvaeqve componens ex ea qvae ante ipsam est obtinetvr nvmeris svbscriptis circvlariter promotis, 1→2→3→11 \rightarrow 2 \rightarrow 3 \rightarrow 1: prima est u2v3−u3v2u_2 v_3 - u_3 v_2; addendo 11 ad qvemqve nvmervm svbscriptvm, 3+13 + 1 pro 11 nvmerando, fit u3v1−u1v3u_3 v_1 - u_1 v_3, qvae est secvnda, et itervm faciendo fit tertia. Vt memoriter scribatvr, adhibentvr vectores basis canonicae, i⃗=(1,0,0)\vec{i} = \left( 1, 0, 0 \right), j⃗=(0,1,0)\vec{j} = \left( 0, 1, 0 \right) et k⃗=(0,0,1)\vec{k} = \left( 0, 0, 1 \right), qvibvs omnis vector scribitvr u⃗=u1i⃗+u2j⃗+u3k⃗\vec{u} = u_1 \vec{i} + u_2 \vec{j} + u_3 \vec{k}, et determinans formalis: tabvla ordinis tertii cvivs primvs versvs vectores i⃗\vec{i}, j⃗\vec{j} et k⃗\vec{k} continet, qvaeqve secvndvm evm versvm evolvitvr sicvt in lectione de determinantibvs.

u⃗×v⃗=∣i⃗j⃗k⃗u1u2u3v1v2v3∣=∣u2u3v2v3∣i⃗−∣u1u3v1v3∣j⃗+∣u1u2v1v2∣k⃗\vec{u} \times \vec{v} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix} = \begin{vmatrix} u_2 & u_3 \\ v_2 & v_3 \end{vmatrix} \vec{i} - \begin{vmatrix} u_1 & u_3 \\ v_1 & v_3 \end{vmatrix} \vec{j} + \begin{vmatrix} u_1 & u_2 \\ v_1 & v_2 \end{vmatrix} \vec{k}

Determinans formalis est qvia eivs elementa non omnia nvmeri svnt, nec vlla proprietas determinantivm ei adhibetvr: formvlam evolvtionis secvndvm primvm versvm tantvm compendio exprimit. Signvm minvs termini secvndi componentem secvndam reddit, qvia −(u1v3−u3v1)=u3v1−u1v3-\left( u_1 v_3 - u_3 v_1 \right) = u_3 v_1 - u_1 v_3. Contra, qvaeqve componens est vervs determinans ordinis secvndi, ex dvabvs colvmnis matricis versvvm u⃗\vec{u} et v⃗\vec{v} formatvs; proprietates capitis «Proprietates algebraicae» in hac observatione nitvntvr.

ExemplvmProdvctvm vectoriale et prodvcta basis canonicae

Compvtare u⃗×v⃗\vec{u} \times \vec{v} pro u⃗=(2,1,0)\vec{u} = \left( 2, 1, 0 \right) et v⃗=(0,2,1)\vec{v} = \left( 0, 2, 1 \right), atqve prodvcta vectorvm basis canonicae inter se.

Demonstratio

  1. ∣u2u3v2v3∣=∣1021∣=1⋅1−0⋅2=1\begin{vmatrix} u_2 & u_3 \\ v_2 & v_3 \end{vmatrix} = \begin{vmatrix} 1 & 0 \\ 2 & 1 \end{vmatrix} = \resaltar{1 \cdot 1 - 0 \cdot 2} = \dato{c1}{1}

    Componens prima: colvmnam primarvm componentivm delemvs et determinantem ordinis secvndi qvi manet compvtamvs.

  2. −∣u1u3v1v3∣=−∣2001∣=−(2−0)=−2-\begin{vmatrix} u_1 & u_3 \\ v_1 & v_3 \end{vmatrix} = -\begin{vmatrix} 2 & 0 \\ 0 & 1 \end{vmatrix} = \resaltar{-\left( 2 - 0 \right)} = \dato{c2}{-2}

    Componens secvnda: colvmnam mediam delemvs et signvm determinantis mvtamvs.

  3. ∣u1u2v1v2∣=∣2102∣=4−0=4\begin{vmatrix} u_1 & u_2 \\ v_1 & v_2 \end{vmatrix} = \begin{vmatrix} 2 & 1 \\ 0 & 2 \end{vmatrix} = \resaltar{4 - 0} = \dato{c3}{4}

    Componens tertia: colvmnam vltimam delemvs.

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

    Tres componentes conivngimvs. Figvra tres vectores ab origine commvni delineat; u⃗\vec{u} in plano horizontali iacet, et u⃗×v⃗\vec{u} \times \vec{v} svrsvm spectat.

  5. i⃗×j⃗=k⃗,j⃗×k⃗=i⃗,k⃗×i⃗=j⃗,j⃗×i⃗=−k⃗\vec{i} \times \vec{j} = \resaltar{\vec{k}}, \quad \vec{j} \times \vec{k} = \resaltar{\vec{i}}, \quad \vec{k} \times \vec{i} = \resaltar{\vec{j}}, \qquad \vec{j} \times \vec{i} = -\vec{k}

    Ex definitione, i⃗×j⃗=(0⋅0−0⋅1,0⋅0−1⋅0,1⋅1−0⋅0)=(0,0,1)\vec{i} \times \vec{j} = \left( 0 \cdot 0 - 0 \cdot 1, 0 \cdot 0 - 1 \cdot 0, 1 \cdot 1 - 0 \cdot 0 \right) = \left( 0, 0, 1 \right), et eodem modo cvm ceteris paribvs. Qvisqve vector basis, mvltiplicatvs per evm qvi ipsvm in ordine circvlari i⃗\vec{i}, j⃗\vec{j}, k⃗\vec{k} excipit, tertivm dat; in ordine contrario, eivs oppositvm.

Proprietates algebraicae

Proprietates prodvcti vectorialis in eivs componentibvs legvntvr. Appellemvs D1D_1, D2D_2 et D3D_3 tres determinantes ordinis secvndi qvi eas formant, colvmnis in ordine circvlari positis: D1D_1 colvmnas secvndam et tertiam matricis versvvm u⃗\vec{u} et v⃗\vec{v} adhibet; D2D_2, tertiam et primam, eo ordine; et D3D_3, primam et secvndam.

u⃗×v⃗=(∣u2u3v2v3∣,∣u3u1v3v1∣,∣u1u2v1v2∣)\vec{u} \times \vec{v} = \left( \begin{vmatrix} u_2 & u_3 \\ v_2 & v_3 \end{vmatrix}, \begin{vmatrix} u_3 & u_1 \\ v_3 & v_1 \end{vmatrix}, \begin{vmatrix} u_1 & u_2 \\ v_1 & v_2 \end{vmatrix} \right)
TheoremaProprietates algebraicae prodvcti vectorialis

Pro omnibvs vectoribvs u⃗\vec{u}, v⃗\vec{v} et w⃗\vec{w} ipsivs R3\mathbb{R}^3 et omnibvs realibvs ss et tt: prodvctvm vectoriale est anticommvtativvm, v⃗×u⃗=−u⃗×v⃗\vec{v} \times \vec{u} = -\vec{u} \times \vec{v}; est bilineare, id est lineare in qvoqve factore, (su⃗+tw⃗)×v⃗=s u⃗×v⃗+t w⃗×v⃗\left( s \vec{u} + t \vec{w} \right) \times \vec{v} = s \, \vec{u} \times \vec{v} + t \, \vec{w} \times \vec{v} et v⃗×(su⃗+tw⃗)=s v⃗×u⃗+t v⃗×w⃗\vec{v} \times \left( s \vec{u} + t \vec{w} \right) = s \, \vec{v} \times \vec{u} + t \, \vec{v} \times \vec{w}; et u⃗×u⃗=0⃗\vec{u} \times \vec{u} = \vec{0}.

Tres proprietates ad eas determinantivm ordinis secvndi redigvntvr, ad qvamqve trivm componentivm adhibitas: lemma de permvtatione dvorvm versvvm et lemma de linearitate in qvoqve versv, in lectione de determinantibvs pro omni ordine demonstrata. Figvra exemplvm «Prodvctvm vectoriale et prodvcta basis canonicae» perseqvitvr.

Demonstratio

  1. (u⃗×v⃗)1=∣u2u3v2v3∣,(u⃗×v⃗)2=∣u3u1v3v1∣,(u⃗×v⃗)3=∣u1u2v1v2∣\left( \vec{u} \times \vec{v} \right)_1 = \begin{vmatrix} \resaltar{u_2} & \resaltar{u_3} \\ v_2 & v_3 \end{vmatrix}, \quad \left( \vec{u} \times \vec{v} \right)_2 = \begin{vmatrix} \resaltar{u_3} & \resaltar{u_1} \\ v_3 & v_1 \end{vmatrix}, \quad \left( \vec{u} \times \vec{v} \right)_3 = \begin{vmatrix} \resaltar{u_1} & \resaltar{u_2} \\ v_1 & v_2 \end{vmatrix}

    Qvaeqve componens DmD_m est determinans matricis ordinis secvndi cvivs primvs versvs ex u⃗\vec{u} et cvivs secvndvs versvs ex v⃗\vec{v} svmitvr, eisdem dvabvs colvmnis.

  2. (v⃗×u⃗)1=∣v2v3u2u3∣=−∣u2u3v2v3∣→v⃗×u⃗=−u⃗×v⃗\left( \vec{v} \times \vec{u} \right)_1 = \begin{vmatrix} \resaltar{v_2} & \resaltar{v_3} \\ \resaltar{u_2} & \resaltar{u_3} \end{vmatrix} = -\begin{vmatrix} u_2 & u_3 \\ v_2 & v_3 \end{vmatrix} \rightarrow \dato{a}{\vec{v} \times \vec{u} = -\vec{u} \times \vec{v}}

    Si u⃗\vec{u} et v⃗\vec{v} permvtantvr, dvo versvs cvivsqve determinantis permvtantvr, et per lemma de permvtatione qvaeqve componens signvm mvtat. In figvra, v⃗×u⃗=(−1,2,−4)\vec{v} \times \vec{u} = \left( -1, 2, -4 \right) deorsvm spectat.

  3. ∣su2+tw2su3+tw3v2v3∣=s∣u2u3v2v3∣+t∣w2w3v2v3∣→(su⃗+tw⃗)×v⃗=s u⃗×v⃗+t w⃗×v⃗\begin{vmatrix} \resaltar{s u_2 + t w_2} & \resaltar{s u_3 + t w_3} \\ v_2 & v_3 \end{vmatrix} = s \begin{vmatrix} u_2 & u_3 \\ v_2 & v_3 \end{vmatrix} + t \begin{vmatrix} w_2 & w_3 \\ v_2 & v_3 \end{vmatrix} \rightarrow \dato{l}{\left( s \vec{u} + t \vec{w} \right) \times \vec{v} = s \, \vec{u} \times \vec{v} + t \, \vec{w} \times \vec{v}}

    Si primvs factor est su⃗+tw⃗s \vec{u} + t \vec{w}, primvs versvs cvivsqve determinantis est ss vicibvs versvs ex u⃗\vec{u} svmptvs plvs tt vicibvs is ex w⃗\vec{w} svmptvs. Per lemma de linearitate, qvaeqve componens eodem modo dividitvr.

  4. v⃗×(su⃗+tw⃗)=−(su⃗+tw⃗)×v⃗=−s u⃗×v⃗−t w⃗×v⃗=s v⃗×u⃗+t v⃗×w⃗\vec{v} \times \left( s \vec{u} + t \vec{w} \right) = \resaltar{-\left( s \vec{u} + t \vec{w} \right) \times \vec{v}} = -s \, \vec{u} \times \vec{v} - t \, \vec{w} \times \vec{v} = \resaltar{s \, \vec{v} \times \vec{u} + t \, \vec{v} \times \vec{w}}

    Linearitas in secvndo factore ex dvabvs proprietatibvs modo demonstratis dedvcitvr: ordo mvtatvr, linearitas in primo factore adhibetvr et ordo rvrsvs mvtatvr.

  5. u⃗×u⃗=(u2u3−u3u2,u3u1−u1u3,u1u2−u2u1)=0⃗\vec{u} \times \vec{u} = \left( \resaltar{u_2 u_3 - u_3 u_2}, \resaltar{u_3 u_1 - u_1 u_3}, \resaltar{u_1 u_2 - u_2 u_1} \right) = \vec{0}

    Si dvo factores congrvvnt, qvisqve determinans dvos versvs aeqvales habet et nvllvs est. Praesertim, i⃗×i⃗=j⃗×j⃗=k⃗×k⃗=0⃗\vec{i} \times \vec{i} = \vec{j} \times \vec{j} = \vec{k} \times \vec{k} = \vec{0}.

Bilinearitate adhibita, prodvctvm vectoriale etiam distribvendo compvtatvr, sicvt prodvctvm svmmarvm, et prodvctis basis canonicae vtendo. Exempli gratia, (i⃗+2j⃗)×(3k⃗)=3 i⃗×k⃗+6 j⃗×k⃗=−3j⃗+6i⃗\left( \vec{i} + 2 \vec{j} \right) \times \left( 3 \vec{k} \right) = 3 \, \vec{i} \times \vec{k} + 6 \, \vec{j} \times \vec{k} = -3 \vec{j} + 6 \vec{i}. Anticommvtativitas dicit ordinem factorvm referre, sicvt in prodvcto matricvm, qvamqvam modo admodvm singvlari: eo inverso, signvm tantvm mvtatvr.

ExemplvmProdvctvm vectoriale non est associativvm

Comparare (i⃗×i⃗)×j⃗\left( \vec{i} \times \vec{i} \right) \times \vec{j} cvm i⃗×(i⃗×j⃗)\vec{i} \times \left( \vec{i} \times \vec{j} \right).

Demonstratio

  1. (i⃗×i⃗)×j⃗=0⃗×j⃗=0⃗\left( \resaltar{\vec{i} \times \vec{i}} \right) \times \vec{j} = \resaltar{\vec{0}} \times \vec{j} = \dato{p}{\vec{0}}

    Per theorema, prodvctvm vectoris per se ipsvm nvllvm est, et per bilinearitatem prodvctvm vectoris nvlli per qvemlibet alivm etiam nvllvm est.

  2. i⃗×(i⃗×j⃗)=i⃗×k⃗\vec{i} \times \left( \resaltar{\vec{i} \times \vec{j}} \right) = \vec{i} \times \resaltar{\vec{k}}

    In altera consociatione, prodvctvm intra parentheses est k⃗\vec{k}, per exemplvm basis canonicae.

  3. i⃗×k⃗=−k⃗×i⃗=−j⃗\vec{i} \times \vec{k} = \resaltar{-\vec{k} \times \vec{i}} = \dato{q}{-\vec{j}}

    Per anticommvtativitatem, i⃗×k⃗\vec{i} \times \vec{k} est oppositvm ipsivs k⃗×i⃗=j⃗\vec{k} \times \vec{i} = \vec{j}.

  4. (i⃗×i⃗)×j⃗=0⃗≠−j⃗=i⃗×(i⃗×j⃗)\left( \vec{i} \times \vec{i} \right) \times \vec{j} = \resaltar{\vec{0}} \neq \resaltar{-\vec{j}} = \vec{i} \times \left( \vec{i} \times \vec{j} \right)

    Dvo effecta diversa svnt: prodvctvm vectoriale non est associativvm, et expressio qvalis u⃗×v⃗×w⃗\vec{u} \times \vec{v} \times \vec{w}, sine parenthesibvs, sensv caret.

Neqve elementvm nevtrvm exstat: nvllvs vector e⃗\vec{e} explet e⃗×u⃗=u⃗\vec{e} \times \vec{u} = \vec{u} pro omni u⃗\vec{u}, qvia e⃗×e⃗=0⃗\vec{e} \times \vec{e} = \vec{0} cogeret e⃗=0⃗\vec{e} = \vec{0}, et tvm e⃗×u⃗=0⃗\vec{e} \times \vec{u} = \vec{0} pro omni u⃗\vec{u}. Prodvctvm vectoriale igitvr nvllam proprietatvm prodvctorvm nvmerorvm habet praeter distribvtivitatem. Capvt «Dvplex prodvctvm vectoriale» regvlam exactam dabit qvae associativitatis loco est.